// https://contest.ucup.ac/contest/1828/problem/9564
pub mod solution {
//{"name":"A. Hey, Have You Seen My Kangaroo?","group":"Universal Cup - The 3rd Universal Cup. Stage 16: Nanjing","url":"https://contest.ucup.ac/contest/1828/problem/9564","interactive":false,"timeLimit":1000,"tests":[{"input":"3 3 6\nULDDRR\n010\n111\n010\n","output":"-1\n4\n2\n1\n0\n0\n0\n0\n0\n"},{"input":"3 3 6\nULDDRR\n010\n111\n011\n","output":"7\n4\n2\n1\n1\n0\n0\n0\n0\n"},{"input":"1 5 1\nR\n11111\n","output":"4\n3\n2\n1\n0\n"}],"testType":"single","input":{"type":"stdin","fileName":null,"pattern":null},"output":{"type":"stdout","fileName":null,"pattern":null},"languages":{"java":{"taskClass":"AHeyHaveYouSeenMyKangaroo"}}}

use crate::algo_lib::collections::bit_set::BitSet;
use crate::algo_lib::collections::fx_hash_map::FxHashMap;
use crate::algo_lib::collections::iter_ext::collect::IterCollect;
use crate::algo_lib::collections::md_arr::arr2d::Arr2dRead;
use crate::algo_lib::collections::min_max::MinimMaxim;
use crate::algo_lib::collections::slice_ext::indices::Indices;
use crate::algo_lib::graph::edges::edge::Edge;
use crate::algo_lib::graph::edges::edge_trait::EdgeTrait;
use crate::algo_lib::graph::Graph;
use crate::algo_lib::io::input::Input;
use crate::algo_lib::io::output::Output;
use crate::algo_lib::misc::recursive_function::Callable;
use crate::algo_lib::misc::recursive_function::RecursiveFunction;
use crate::algo_lib::misc::test_type::TaskType;
use crate::algo_lib::misc::test_type::TestType;
use crate::algo_lib::string::str::StrReader;
use std::cmp::Reverse;

type PreCalc = ();

fn solve(input: &mut Input, out: &mut Output, _test_case: usize, _data: &mut PreCalc) {
    let n = input.read_size();
    let m = input.read_size();
    let k = input.read_size();
    let s = input.read_str();
    let a = input.read_char_table(n, m);

    let mut graph = Graph::new(n * m);
    let mut is_root = BitSet::new(n * m);
    let mut x = vec![None; n * m];

    let mv = |r: usize, c: usize, cc: u8| -> (usize, usize) {
        match cc {
            b'U' => {
                if r > 0 && a[(r - 1, c)] == b'1' {
                    return (r - 1, c);
                }
            }
            b'D' => {
                if r + 1 < n && a[(r + 1, c)] == b'1' {
                    return (r + 1, c);
                }
            }
            b'L' => {
                if c > 0 && a[(r, c - 1)] == b'1' {
                    return (r, c - 1);
                }
            }
            b'R' => {
                if c + 1 < m && a[(r, c + 1)] == b'1' {
                    return (r, c + 1);
                }
            }
            _ => unreachable!(),
        }
        (r, c)
    };
    for i in 0..n {
        for j in 0..m {
            if a[(i, j)] == b'0' {
                continue;
            }
            let mut r = i;
            let mut c = j;
            for cc in s.iter() {
                (r, c) = mv(r, c, cc);
            }
            x[i * m + j] = Some(r * m + c);
        }
    }
    let mut done = vec![0; n * m];
    let mut root_shift = (0..n * m).collect_vec();
    for i in 0..n * m {
        if done[i] == 2 || x[i].is_none() {
            continue;
        }
        let mut cur = i;
        while done[cur] == 0 {
            done[cur] = 1;
            cur = x[cur].unwrap();
        }
        while done[cur] == 1 {
            is_root.set(cur);
            done[cur] = 2;
            root_shift[x[cur].unwrap()] = cur;
            cur = x[cur].unwrap();
        }
        cur = i;
        while done[cur] == 1 {
            done[cur] = 2;
            cur = x[cur].unwrap();
        }
    }
    for i in 0..n * m {
        if let Some(to) = x[i] {
            if !is_root[i] {
                graph.add_edge(Edge::new(root_shift[to], i));
            }
        }
    }
    let mut joins = Vec::new();
    let mut ans = Vec::with_capacity(n * m);
    for i in is_root.iter() {
        let mut rec = RecursiveFunction::new(|rec, vert: usize| -> usize {
            let mut lens = Vec::new();
            let mut pos = Vec::new();
            for e in &graph[vert] {
                lens.push(rec.call(e.to()) + 1);
                pos.push((e.to() / m, e.to() % m));
            }
            if is_root[vert] {
                lens.push(usize::MAX);
                pos.push((vert / m, vert % m));
            }
            let mut good = BitSet::new(pos.len());
            good.fill(true);
            for i in 0..k {
                let mut seen = FxHashMap::<_, usize>::default();
                for j in pos.indices() {
                    if !good[j] {
                        continue;
                    }
                    pos[j] = mv(pos[j].0, pos[j].1, s[i]);
                    if let Some(&o) = seen.get(&pos[j]) {
                        for x in 0..lens[o].min(lens[j]) {
                            joins.push(i + 1 + x * k);
                        }
                        let cand = lens[j];
                        lens[o].maxim(cand);
                        good.unset(j);
                    } else {
                        seen.insert(pos[j], j);
                    }
                }
            }
            lens.get(0).copied().unwrap_or(0)
        });
        rec.call(i);
        ans.push(None);
    }
    ans.pop();
    joins.sort_by_key(|&x| Reverse(x));
    ans.extend(joins.into_iter().map(Some));
    ans.resize(n * m, Some(0));
    out.print_per_line(&ans);
}

pub static TEST_TYPE: TestType = TestType::Single;
pub static TASK_TYPE: TaskType = TaskType::Classic;

pub(crate) fn run(mut input: Input, mut output: Output) -> bool {
    let mut pre_calc = ();

    match TEST_TYPE {
        TestType::Single => solve(&mut input, &mut output, 1, &mut pre_calc),
        TestType::MultiNumber => {
            let t = input.read();
            for i in 1..=t {
                solve(&mut input, &mut output, i, &mut pre_calc);
            }
        }
        TestType::MultiEof => {
            let mut i = 1;
            while input.peek().is_some() {
                solve(&mut input, &mut output, i, &mut pre_calc);
                i += 1;
            }
        }
    }
    output.flush();
    match TASK_TYPE {
        TaskType::Classic => input.is_empty(),
        TaskType::Interactive => true,
    }
}

}
pub mod algo_lib {
#![allow(clippy::too_many_arguments)]
#![allow(clippy::type_complexity)]
#![allow(clippy::missing_safety_doc)]
#![allow(clippy::legacy_numeric_constants)]

pub mod collections {
pub mod bit_set {
use crate::algo_lib::collections::slice_ext::legacy_fill::LegacyFill;
use crate::algo_lib::numbers::num_traits::bit_ops::BitOps;
use std::ops::BitAndAssign;
use std::ops::BitOrAssign;
use std::ops::Index;
use std::ops::ShlAssign;
use std::ops::ShrAssign;

const TRUE: bool = true;
const FALSE: bool = false;

#[derive(Clone, Eq, PartialEq, Hash)]
pub struct BitSet {
    data: Vec<u64>,
    len: usize,
}

impl BitSet {
    pub fn new(len: usize) -> Self {
        let data_len = if len == 0 {
            0
        } else {
            Self::index(len - 1) + 1
        };
        Self {
            data: vec![0; data_len],
            len,
        }
    }

    pub fn from_slice(len: usize, set: &[usize]) -> Self {
        let mut res = Self::new(len);
        for &i in set {
            res.set(i);
        }
        res
    }

    pub fn set(&mut self, at: usize) {
        assert!(at < self.len);
        self.data[Self::index(at)].set_bit(at & 63);
    }

    pub fn unset(&mut self, at: usize) {
        assert!(at < self.len);
        self.data[Self::index(at)].unset_bit(at & 63);
    }

    pub fn change(&mut self, at: usize, value: bool) {
        if value {
            self.set(at);
        } else {
            self.unset(at);
        }
    }

    pub fn flip(&mut self, at: usize) {
        self.change(at, !self[at]);
    }

    #[allow(clippy::len_without_is_empty)]
    pub fn len(&self) -> usize {
        self.len
    }

    pub fn fill(&mut self, value: bool) {
        // 1.43
        self.data.legacy_fill(if value { std::u64::MAX } else { 0 });
        if value {
            self.fix_last();
        }
    }

    pub fn is_superset(&self, other: &Self) -> bool {
        assert_eq!(self.len, other.len);
        for i in 0..self.data.len() {
            if self.data[i] & other.data[i] != other.data[i] {
                return false;
            }
        }
        true
    }

    pub fn is_subset(&self, other: &Self) -> bool {
        other.is_superset(self)
    }

    pub fn iter(&self) -> impl Iterator<Item = usize> + '_ {
        self.into_iter()
    }

    fn index(at: usize) -> usize {
        at >> 6
    }

    pub fn count_ones(&self) -> usize {
        self.data.iter().map(|x| x.count_ones() as usize).sum()
    }

    fn fix_last(&mut self) {
        if self.len & 63 != 0 {
            let mask = (1 << (self.len & 63)) - 1;
            *self.data.last_mut().unwrap() &= mask;
        }
    }
}

pub struct BitSetIter<'s> {
    at: usize,
    inside: usize,
    set: &'s BitSet,
}

impl<'s> Iterator for BitSetIter<'s> {
    type Item = usize;

    fn next(&mut self) -> Option<Self::Item> {
        while self.at < self.set.data.len()
            && (self.inside == 64 || (self.set.data[self.at] >> self.inside) == 0)
        {
            self.at += 1;
            self.inside = 0;
        }
        if self.at == self.set.data.len() {
            None
        } else {
            while !self.set.data[self.at].is_set(self.inside) {
                self.inside += 1;
            }
            let res = self.at * 64 + self.inside;
            if res < self.set.len {
                self.inside += 1;
                Some(res)
            } else {
                None
            }
        }
    }
}

impl<'a> IntoIterator for &'a BitSet {
    type Item = usize;
    type IntoIter = BitSetIter<'a>;

    fn into_iter(self) -> Self::IntoIter {
        BitSetIter {
            at: 0,
            inside: 0,
            set: self,
        }
    }
}

impl BitOrAssign<&BitSet> for BitSet {
    fn bitor_assign(&mut self, rhs: &BitSet) {
        assert_eq!(self.len, rhs.len);
        for (i, &j) in self.data.iter_mut().zip(rhs.data.iter()) {
            *i |= j;
        }
    }
}

impl BitAndAssign<&BitSet> for BitSet {
    fn bitand_assign(&mut self, rhs: &BitSet) {
        assert_eq!(self.len, rhs.len);
        for (i, &j) in self.data.iter_mut().zip(rhs.data.iter()) {
            *i &= j;
        }
    }
}

impl ShlAssign<usize> for BitSet {
    fn shl_assign(&mut self, rhs: usize) {
        if rhs == 0 {
            return;
        }
        let small_shift = rhs & 63;
        if small_shift != 0 {
            let mut carry = 0;
            for i in 0..self.data.len() {
                let new_carry = self.data[i] >> (64 - small_shift);
                self.data[i] <<= small_shift;
                self.data[i] |= carry;
                carry = new_carry;
            }
        }
        let big_shift = rhs >> 6;
        if big_shift != 0 {
            self.data.rotate_right(big_shift);
            self.data[..big_shift].fill(0);
        }
        self.fix_last();
    }
}

impl ShrAssign<usize> for BitSet {
    fn shr_assign(&mut self, rhs: usize) {
        if rhs == 0 {
            return;
        }
        let small_shift = rhs & 63;
        if small_shift != 0 {
            let mut carry = 0;
            for i in (0..self.data.len()).rev() {
                let new_carry = self.data[i] << (64 - small_shift);
                self.data[i] >>= small_shift;
                self.data[i] |= carry;
                carry = new_carry;
            }
        }
        let big_shift = rhs >> 6;
        if big_shift != 0 {
            self.data.rotate_left(big_shift);
            let from = self.data.len() - big_shift;
            self.data[from..].fill(0);
        }
    }
}

impl Index<usize> for BitSet {
    type Output = bool;

    fn index(&self, at: usize) -> &Self::Output {
        assert!(at < self.len);
        if self.data[Self::index(at)].is_set(at & 63) {
            &TRUE
        } else {
            &FALSE
        }
    }
}

impl From<Vec<bool>> for BitSet {
    fn from(data: Vec<bool>) -> Self {
        let mut res = Self::new(data.len());
        for (i, &value) in data.iter().enumerate() {
            res.change(i, value);
        }
        res
    }
}
}
pub mod dsu {
use crate::algo_lib::collections::iter_ext::collect::IterCollect;
use crate::algo_lib::collections::slice_ext::bounds::Bounds;
use crate::algo_lib::collections::slice_ext::legacy_fill::LegacyFill;
use std::cell::Cell;

#[derive(Clone)]
pub struct DSU {
    id: Vec<Cell<u32>>,
    size: Vec<u32>,
    count: usize,
}

impl DSU {
    pub fn new(n: usize) -> Self {
        Self {
            id: (0..n).map(|i| Cell::new(i as u32)).collect_vec(),
            size: vec![1; n],
            count: n,
        }
    }

    pub fn size(&self, i: usize) -> usize {
        self.size[self.get(i)] as usize
    }

    #[allow(clippy::len_without_is_empty)]
    pub fn len(&self) -> usize {
        self.id.len()
    }

    pub fn iter(&self) -> impl Iterator<Item = usize> + '_ {
        self.id.iter().enumerate().filter_map(|(i, id)| {
            if (i as u32) == id.get() {
                Some(i)
            } else {
                None
            }
        })
    }

    pub fn set_count(&self) -> usize {
        self.count
    }

    pub fn join(&mut self, mut a: usize, mut b: usize) -> bool {
        a = self.get(a);
        b = self.get(b);
        if a == b {
            false
        } else {
            self.size[a] += self.size[b];
            self.id[b].replace(a as u32);
            self.count -= 1;
            true
        }
    }

    pub fn get(&self, i: usize) -> usize {
        if self.id[i].get() != i as u32 {
            let res = self.get(self.id[i].get() as usize);
            self.id[i].replace(res as u32);
        }
        self.id[i].get() as usize
    }

    pub fn clear(&mut self) {
        self.count = self.id.len();
        self.size.legacy_fill(1);
        self.id.iter().enumerate().for_each(|(i, id)| {
            id.replace(i as u32);
        });
    }

    pub fn parts(&self) -> Vec<Vec<usize>> {
        let roots = self.iter().collect_vec();
        let mut res = vec![Vec::new(); roots.len()];
        for i in 0..self.id.len() {
            res[roots.as_slice().bin_search(&self.get(i)).unwrap()].push(i);
        }
        res
    }
}
}
pub mod fx_hash_map {
// Copyright 2015 The Rust Project Developers. See the COPYRIGHT at
// http://rust-lang.org/COPYRIGHT.
//
// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
// <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your
// option. This file may not be copied, modified, or distributed
// except according to those terms.

use std::cell::Cell;
use std::convert::TryInto;
use std::time::SystemTime;
use std::collections::HashMap;
use std::collections::HashSet;
use std::hash::BuildHasherDefault;
use std::hash::Hasher;
use std::mem::size_of;
use std::ops::BitXor;

pub type FxHashMap<K, V> = HashMap<K, V, BuildHasherDefault<FxHasher>>;

pub type FxHashSet<V> = HashSet<V, BuildHasherDefault<FxHasher>>;

#[derive(Default)]
pub struct FxHasher {
    hash: usize,
}

thread_local! {
    static K: Cell<usize> = Cell::new(
        ((SystemTime::UNIX_EPOCH.elapsed().unwrap().as_nanos().wrapping_mul(2) + 1) & 0xFFFFFFFFFFFFFFFF) as usize
    );
}

impl FxHasher {
    #[inline]
    fn add_to_hash(&mut self, i: usize) {
        self.hash = self
            .hash
            .rotate_left(5)
            .bitxor(i)
            .wrapping_mul(K.with(|k| k.get()));
    }
}

impl Hasher for FxHasher {
    #[inline]
    fn write(&mut self, mut bytes: &[u8]) {
        let read_usize = |bytes: &[u8]| u64::from_ne_bytes(bytes[..8].try_into().unwrap());

        let mut hash = FxHasher { hash: self.hash };
        while bytes.len() >= size_of::<usize>() {
            hash.add_to_hash(read_usize(bytes) as usize);
            bytes = &bytes[size_of::<usize>()..];
        }
        if (size_of::<usize>() > 4) && (bytes.len() >= 4) {
            hash.add_to_hash(u32::from_ne_bytes(bytes[..4].try_into().unwrap()) as usize);
            bytes = &bytes[4..];
        }
        if (size_of::<usize>() > 2) && bytes.len() >= 2 {
            hash.add_to_hash(u16::from_ne_bytes(bytes[..2].try_into().unwrap()) as usize);
            bytes = &bytes[2..];
        }
        if (size_of::<usize>() > 1) && !bytes.is_empty() {
            hash.add_to_hash(bytes[0] as usize);
        }
        self.hash = hash.hash;
    }

    #[inline]
    fn write_u8(&mut self, i: u8) {
        self.add_to_hash(i as usize);
    }

    #[inline]
    fn write_u16(&mut self, i: u16) {
        self.add_to_hash(i as usize);
    }

    #[inline]
    fn write_u32(&mut self, i: u32) {
        self.add_to_hash(i as usize);
    }

    #[inline]
    fn write_u64(&mut self, i: u64) {
        self.add_to_hash(i as usize);
    }

    #[inline]
    fn write_usize(&mut self, i: usize) {
        self.add_to_hash(i);
    }

    #[inline]
    fn finish(&self) -> u64 {
        self.hash as u64
    }
}
}
pub mod iter_ext {
pub mod collect {
pub trait IterCollect<T>: Iterator<Item = T> + Sized {
    fn collect_vec(self) -> Vec<T> {
        self.collect()
    }
}

impl<T, I: Iterator<Item = T> + Sized> IterCollect<T> for I {}
}
}
pub mod md_arr {
pub mod arr2d {
use crate::algo_lib::collections::slice_ext::legacy_fill::LegacyFill;
use crate::algo_lib::io::input::Input;
use crate::algo_lib::io::input::Readable;
use crate::algo_lib::io::output::Output;
use crate::algo_lib::io::output::Writable;
use std::mem::MaybeUninit;
use std::ops::Index;
use std::ops::IndexMut;
use std::ops::Range;
use std::slice::Iter;
use std::vec::IntoIter;

#[derive(Clone, Eq, PartialEq, Default)]
pub struct Arr2d<T> {
    d1: usize,
    d2: usize,
    data: Vec<T>,
}

impl<T: Clone> Arr2d<T> {
    pub fn new(d1: usize, d2: usize, value: T) -> Self {
        Self {
            d1,
            d2,
            data: vec![value; d1 * d2],
        }
    }
}

impl<T> Arr2d<T> {
    pub fn generate<F>(d1: usize, d2: usize, mut gen: F) -> Self
    where
        F: FnMut(usize, usize) -> T,
    {
        let mut data = Vec::with_capacity(d1 * d2);
        for i in 0usize..d1 {
            for j in 0usize..d2 {
                data.push(gen(i, j));
            }
        }
        Self { d1, d2, data }
    }

    pub fn d1(&self) -> usize {
        self.d1
    }

    pub fn d2(&self) -> usize {
        self.d2
    }

    pub fn iter(&self) -> Iter<'_, T> {
        self.data.iter()
    }

    pub fn iter_mut(&mut self) -> impl Iterator<Item = &mut T> {
        self.data.iter_mut()
    }

    pub fn row(&self, row: usize) -> impl Iterator<Item = &T> {
        assert!(row < self.d1);
        self.data.iter().skip(row * self.d2).take(self.d2)
    }

    pub fn row_mut(&mut self, row: usize) -> impl Iterator<Item = &mut T> {
        assert!(row < self.d1);
        self.data.iter_mut().skip(row * self.d2).take(self.d2)
    }

    pub fn column(&self, col: usize) -> impl Iterator<Item = &T> {
        assert!(col < self.d2);
        self.data.iter().skip(col).step_by(self.d2)
    }

    pub fn column_mut(&mut self, col: usize) -> impl Iterator<Item = &mut T> {
        assert!(col < self.d2);
        self.data.iter_mut().skip(col).step_by(self.d2)
    }

    pub fn swap(&mut self, r1: usize, c1: usize, r2: usize, c2: usize) {
        assert!(r1 < self.d1);
        assert!(r2 < self.d1);
        assert!(c1 < self.d2);
        assert!(c2 < self.d2);
        self.data.swap(r1 * self.d2 + c1, r2 * self.d2 + c2);
    }

    pub fn rows(&self) -> Range<usize> {
        0..self.d1
    }

    pub fn cols(&self) -> Range<usize> {
        0..self.d2
    }

    pub fn swap_rows(&mut self, r1: usize, r2: usize) {
        assert!(r1 < self.d1);
        assert!(r2 < self.d1);
        if r1 == r2 {
            return;
        }
        let (r1, r2) = (r1.min(r2), r1.max(r2));
        let (head, tail) = self.data.split_at_mut(r2 * self.d2);
        head[r1 * self.d2..(r1 + 1) * self.d2].swap_with_slice(&mut tail[..self.d2]);
    }

    pub fn rotate_clockwise(self) -> Self {
        unsafe {
            let d1 = self.d1;
            let d2 = self.d2;
            let mut res = MaybeUninit::new(Vec::with_capacity(d1 * d2));
            (*res.as_mut_ptr()).set_len(d1 * d2);
            for (id, element) in self.into_iter().enumerate() {
                let (i, j) = (id / d2, id % d2);
                let ptr: *mut T = (*res.as_mut_ptr()).as_mut_ptr();
                ptr.add(j * d1 + d1 - i - 1).write(element);
            }
            Self {
                d1: d2,
                d2: d1,
                data: res.assume_init(),
            }
        }
    }

    pub fn rotate_counterclockwise(self) -> Self {
        unsafe {
            let d1 = self.d1;
            let d2 = self.d2;
            let mut res = MaybeUninit::new(Vec::with_capacity(d1 * d2));
            (*res.as_mut_ptr()).set_len(d1 * d2);
            for (id, element) in self.into_iter().enumerate() {
                let (i, j) = (id / d2, id % d2);
                let ptr: *mut T = (*res.as_mut_ptr()).as_mut_ptr();
                ptr.add((d2 - j - 1) * d1 + i).write(element);
            }
            Self {
                d1: d2,
                d2: d1,
                data: res.assume_init(),
            }
        }
    }
}

impl<T: Clone> Arr2d<T> {
    pub fn fill(&mut self, elem: T) {
        self.data.legacy_fill(elem);
    }

    pub fn transpose(&self) -> Self {
        Self::generate(self.d2, self.d1, |i, j| self[(j, i)].clone())
    }
}

impl<T> Index<(usize, usize)> for Arr2d<T> {
    type Output = T;

    fn index(&self, (row, col): (usize, usize)) -> &Self::Output {
        assert!(row < self.d1);
        assert!(col < self.d2);
        &self.data[self.d2 * row + col]
    }
}

impl<T> Index<usize> for Arr2d<T> {
    type Output = [T];

    fn index(&self, index: usize) -> &Self::Output {
        &self.data[self.d2 * index..self.d2 * (index + 1)]
    }
}

impl<T> IndexMut<(usize, usize)> for Arr2d<T> {
    fn index_mut(&mut self, (row, col): (usize, usize)) -> &mut T {
        assert!(row < self.d1);
        assert!(col < self.d2);
        &mut self.data[self.d2 * row + col]
    }
}

impl<T> IndexMut<usize> for Arr2d<T> {
    fn index_mut(&mut self, index: usize) -> &mut [T] {
        &mut self.data[self.d2 * index..self.d2 * (index + 1)]
    }
}

impl<T> AsRef<Vec<T>> for Arr2d<T> {
    fn as_ref(&self) -> &Vec<T> {
        &self.data
    }
}

impl<T> AsMut<Vec<T>> for Arr2d<T> {
    fn as_mut(&mut self) -> &mut Vec<T> {
        &mut self.data
    }
}

impl<T: Writable> Writable for Arr2d<T> {
    fn write(&self, output: &mut Output) {
        let mut at = 0usize;
        for i in 0usize..self.d1 {
            if i != 0 {
                output.put(b'\n');
            }
            for j in 0usize..self.d2 {
                if j != 0 {
                    output.put(b' ');
                }
                self.data[at].write(output);
                at += 1;
            }
        }
    }
}

impl<T> IntoIterator for Arr2d<T> {
    type Item = T;
    type IntoIter = IntoIter<T>;

    fn into_iter(self) -> Self::IntoIter {
        self.data.into_iter()
    }
}

impl<'a, T> IntoIterator for &'a Arr2d<T> {
    type Item = &'a T;
    type IntoIter = Iter<'a, T>;

    fn into_iter(self) -> Self::IntoIter {
        self.iter()
    }
}

pub trait Arr2dRead {
    fn read_table<T: Readable>(&mut self, d1: usize, d2: usize) -> Arr2d<T>;
    fn read_int_table(&mut self, d1: usize, d2: usize) -> Arr2d<i32>;
    fn read_long_table(&mut self, d1: usize, d2: usize) -> Arr2d<i64>;
    fn read_size_table(&mut self, d1: usize, d2: usize) -> Arr2d<usize>;
    fn read_char_table(&mut self, d1: usize, d2: usize) -> Arr2d<u8>;
}

impl Arr2dRead for Input<'_> {
    fn read_table<T: Readable>(&mut self, d1: usize, d2: usize) -> Arr2d<T> {
        Arr2d::generate(d1, d2, |_, _| self.read())
    }

    fn read_int_table(&mut self, d1: usize, d2: usize) -> Arr2d<i32> {
        self.read_table(d1, d2)
    }

    fn read_long_table(&mut self, d1: usize, d2: usize) -> Arr2d<i64> {
        self.read_table(d1, d2)
    }

    fn read_size_table(&mut self, d1: usize, d2: usize) -> Arr2d<usize> {
        self.read_table(d1, d2)
    }

    fn read_char_table(&mut self, d1: usize, d2: usize) -> Arr2d<u8> {
        self.read_table(d1, d2)
    }
}

pub trait Arr2dCharWrite {
    fn print_table(&mut self, table: &Arr2d<u8>);
}

impl Arr2dCharWrite for Output<'_> {
    fn print_table(&mut self, table: &Arr2d<u8>) {
        let mut at = 0usize;
        for _ in 0..table.d1 {
            for _ in 0..table.d2 {
                self.put(table.data[at]);
                at += 1;
            }
            self.put(b'\n');
        }
        self.maybe_flush();
    }
}

impl<T: Readable> Readable for Arr2d<T> {
    fn read(input: &mut Input) -> Self {
        let d1 = input.read();
        let d2 = input.read();
        input.read_table(d1, d2)
    }
}
}
}
pub mod min_max {
pub trait MinimMaxim<Rhs = Self>: PartialOrd + Sized {
    fn minim(&mut self, other: Rhs) -> bool;

    fn maxim(&mut self, other: Rhs) -> bool;
}

impl<T: PartialOrd> MinimMaxim for T {
    fn minim(&mut self, other: Self) -> bool {
        if other < *self {
            *self = other;
            true
        } else {
            false
        }
    }

    fn maxim(&mut self, other: Self) -> bool {
        if other > *self {
            *self = other;
            true
        } else {
            false
        }
    }
}

impl<T: PartialOrd> MinimMaxim<T> for Option<T> {
    fn minim(&mut self, other: T) -> bool {
        match self {
            None => {
                *self = Some(other);
                true
            }
            Some(v) => v.minim(other),
        }
    }

    fn maxim(&mut self, other: T) -> bool {
        match self {
            None => {
                *self = Some(other);
                true
            }
            Some(v) => v.maxim(other),
        }
    }
}
}
pub mod slice_ext {
pub mod backward {
use std::ops::Index;
use std::ops::IndexMut;

pub struct Back(pub usize);

impl<T> Index<Back> for [T] {
    type Output = T;

    fn index(&self, index: Back) -> &Self::Output {
        &self[self.len() - index.0 - 1]
    }
}

impl<T> IndexMut<Back> for [T] {
    fn index_mut(&mut self, index: Back) -> &mut Self::Output {
        &mut self[self.len() - index.0 - 1]
    }
}

impl<T> Index<Back> for Vec<T> {
    type Output = T;

    fn index(&self, index: Back) -> &Self::Output {
        self.as_slice().index(index)
    }
}

impl<T> IndexMut<Back> for Vec<T> {
    fn index_mut(&mut self, index: Back) -> &mut Self::Output {
        self.as_mut_slice().index_mut(index)
    }
}
}
pub mod bounds {
pub trait Bounds<T: PartialOrd> {
    fn lower_bound(&self, el: &T) -> usize;
    fn upper_bound(&self, el: &T) -> usize;
    fn bin_search(&self, el: &T) -> Option<usize>;
    fn more(&self, el: &T) -> usize;
    fn more_or_eq(&self, el: &T) -> usize;
    fn less(&self, el: &T) -> usize;
    fn less_or_eq(&self, el: &T) -> usize;
}

impl<T: PartialOrd> Bounds<T> for [T] {
    fn lower_bound(&self, el: &T) -> usize {
        let mut left = 0;
        let mut right = self.len();
        while left < right {
            let mid = left + ((right - left) >> 1);
            if &self[mid] < el {
                left = mid + 1;
            } else {
                right = mid;
            }
        }
        left
    }

    fn upper_bound(&self, el: &T) -> usize {
        let mut left = 0;
        let mut right = self.len();
        while left < right {
            let mid = left + ((right - left) >> 1);
            if &self[mid] <= el {
                left = mid + 1;
            } else {
                right = mid;
            }
        }
        left
    }

    fn bin_search(&self, el: &T) -> Option<usize> {
        let at = self.lower_bound(el);
        if at == self.len() || &self[at] != el {
            None
        } else {
            Some(at)
        }
    }

    fn more(&self, el: &T) -> usize {
        self.len() - self.upper_bound(el)
    }

    fn more_or_eq(&self, el: &T) -> usize {
        self.len() - self.lower_bound(el)
    }

    fn less(&self, el: &T) -> usize {
        self.lower_bound(el)
    }

    fn less_or_eq(&self, el: &T) -> usize {
        self.upper_bound(el)
    }
}
}
pub mod indices {
use std::ops::Range;

pub trait Indices {
    fn indices(&self) -> Range<usize>;
}

impl<T> Indices for [T] {
    fn indices(&self) -> Range<usize> {
        0..self.len()
    }
}
}
pub mod legacy_fill {
// 1.50
pub trait LegacyFill<T> {
    fn legacy_fill(&mut self, val: T);
}

impl<T: Clone> LegacyFill<T> for [T] {
    fn legacy_fill(&mut self, val: T) {
        for el in self.iter_mut() {
            *el = val.clone();
        }
    }
}
}
}
pub mod vec_ext {
pub mod default {
pub fn default_vec<T: Default>(len: usize) -> Vec<T> {
    let mut v = Vec::with_capacity(len);
    for _ in 0..len {
        v.push(T::default());
    }
    v
}
}
}
}
pub mod graph {
use crate::algo_lib::collections::dsu::DSU;
use crate::algo_lib::graph::edges::bi_edge::BiEdge;
use crate::algo_lib::graph::edges::edge::Edge;
use crate::algo_lib::graph::edges::edge_trait::BidirectionalEdgeTrait;
use crate::algo_lib::graph::edges::edge_trait::EdgeTrait;
use std::ops::Index;
use std::ops::IndexMut;


#[derive(Clone)]
pub struct Graph<E: EdgeTrait> {
    edges: Vec<Vec<E>>,
    edge_count: usize,
}

impl<E: EdgeTrait> Graph<E> {
    pub fn new(vertex_count: usize) -> Self {
        Self {
            edges: vec![Vec::new(); vertex_count],
            edge_count: 0,
        }
    }

    pub fn add_edge(&mut self, (from, mut edge): (usize, E)) -> usize {
        let to = edge.to();
        assert!(to < self.vertex_count());
        let direct_id = self.edges[from].len();
        edge.set_id(self.edge_count);
        self.edges[from].push(edge);
        if E::REVERSABLE {
            let rev_id = self.edges[to].len();
            self.edges[from][direct_id].set_reverse_id(rev_id);
            let mut rev_edge = self.edges[from][direct_id].reverse_edge(from);
            rev_edge.set_id(self.edge_count);
            rev_edge.set_reverse_id(direct_id);
            self.edges[to].push(rev_edge);
        }
        self.edge_count += 1;
        direct_id
    }

    pub fn add_vertices(&mut self, cnt: usize) {
        self.edges.resize(self.edges.len() + cnt, Vec::new());
    }

    pub fn clear(&mut self) {
        self.edge_count = 0;
        for ve in self.edges.iter_mut() {
            ve.clear();
        }
    }

    pub fn vertex_count(&self) -> usize {
        self.edges.len()
    }

    pub fn edge_count(&self) -> usize {
        self.edge_count
    }

    pub fn degrees(&self) -> Vec<usize> {
        self.edges.iter().map(|v| v.len()).collect()
    }
}

impl<E: BidirectionalEdgeTrait> Graph<E> {
    pub fn is_tree(&self) -> bool {
        if self.edge_count + 1 != self.vertex_count() {
            false
        } else {
            self.is_connected()
        }
    }

    pub fn is_forest(&self) -> bool {
        let mut dsu = DSU::new(self.vertex_count());
        for i in 0..self.vertex_count() {
            for e in self[i].iter() {
                if i <= e.to() && !dsu.join(i, e.to()) {
                    return false;
                }
            }
        }
        true
    }

    pub fn is_connected(&self) -> bool {
        let mut dsu = DSU::new(self.vertex_count());
        for i in 0..self.vertex_count() {
            for e in self[i].iter() {
                dsu.join(i, e.to());
            }
        }
        dsu.set_count() == 1
    }
}

impl<E: EdgeTrait> Index<usize> for Graph<E> {
    type Output = [E];

    fn index(&self, index: usize) -> &Self::Output {
        &self.edges[index]
    }
}

impl<E: EdgeTrait> IndexMut<usize> for Graph<E> {
    fn index_mut(&mut self, index: usize) -> &mut Self::Output {
        &mut self.edges[index]
    }
}

impl Graph<Edge<()>> {
    pub fn from_edges(n: usize, edges: &[(usize, usize)]) -> Self {
        let mut graph = Self::new(n);
        for &(from, to) in edges {
            graph.add_edge(Edge::new(from, to));
        }
        graph
    }
}

impl<P: Clone> Graph<Edge<P>> {
    pub fn from_edges_with_payload(n: usize, edges: &[(usize, usize, P)]) -> Self {
        let mut graph = Self::new(n);
        for (from, to, p) in edges.iter() {
            graph.add_edge(Edge::with_payload(*from, *to, p.clone()));
        }
        graph
    }
}

impl Graph<BiEdge<()>> {
    pub fn from_biedges(n: usize, edges: &[(usize, usize)]) -> Self {
        let mut graph = Self::new(n);
        for &(from, to) in edges {
            graph.add_edge(BiEdge::new(from, to));
        }
        graph
    }
}

impl<P: Clone> Graph<BiEdge<P>> {
    pub fn from_biedges_with_payload(n: usize, edges: &[(usize, usize, P)]) -> Self {
        let mut graph = Self::new(n);
        for (from, to, p) in edges.iter() {
            graph.add_edge(BiEdge::with_payload(*from, *to, p.clone()));
        }
        graph
    }
}
pub mod edges {
pub mod bi_edge {
use crate::algo_lib::graph::edges::bi_edge_trait::BiEdgeTrait;
use crate::algo_lib::graph::edges::edge_id::EdgeId;
use crate::algo_lib::graph::edges::edge_id::NoId;
use crate::algo_lib::graph::edges::edge_id::WithId;
use crate::algo_lib::graph::edges::edge_trait::BidirectionalEdgeTrait;
use crate::algo_lib::graph::edges::edge_trait::EdgeTrait;

#[derive(Clone)]
pub struct BiEdgeRaw<Id: EdgeId, P> {
    to: u32,
    id: Id,
    payload: P,
}

impl<Id: EdgeId> BiEdgeRaw<Id, ()> {
    pub fn new(from: usize, to: usize) -> (usize, Self) {
        (
            from,
            Self {
                to: to as u32,
                id: Id::new(),
                payload: (),
            },
        )
    }
}

impl<Id: EdgeId, P> BiEdgeRaw<Id, P> {
    pub fn with_payload(from: usize, to: usize, payload: P) -> (usize, Self) {
        (from, Self::with_payload_impl(to, payload))
    }

    fn with_payload_impl(to: usize, payload: P) -> BiEdgeRaw<Id, P> {
        Self {
            to: to as u32,
            id: Id::new(),
            payload,
        }
    }
}

impl<Id: EdgeId, P: Clone> BidirectionalEdgeTrait for BiEdgeRaw<Id, P> {}

impl<Id: EdgeId, P: Clone> EdgeTrait for BiEdgeRaw<Id, P> {
    type Payload = P;

    const REVERSABLE: bool = true;

    fn to(&self) -> usize {
        self.to as usize
    }

    fn id(&self) -> usize {
        self.id.id()
    }

    fn set_id(&mut self, id: usize) {
        self.id.set_id(id);
    }

    fn reverse_id(&self) -> usize {
        panic!("no reverse id")
    }

    fn set_reverse_id(&mut self, _: usize) {}

    fn reverse_edge(&self, from: usize) -> Self {
        Self::with_payload_impl(from, self.payload.clone())
    }

    fn payload(&self) -> &P {
        &self.payload
    }
}

impl<Id: EdgeId, P: Clone> BiEdgeTrait for BiEdgeRaw<Id, P> {}

pub type BiEdge<P> = BiEdgeRaw<NoId, P>;
pub type BiEdgeWithId<P> = BiEdgeRaw<WithId, P>;
}
pub mod bi_edge_trait {
use crate::algo_lib::graph::edges::edge_trait::EdgeTrait;

pub trait BiEdgeTrait: EdgeTrait {}
}
pub mod edge {
use crate::algo_lib::graph::edges::edge_id::EdgeId;
use crate::algo_lib::graph::edges::edge_id::NoId;
use crate::algo_lib::graph::edges::edge_id::WithId;
use crate::algo_lib::graph::edges::edge_trait::EdgeTrait;

#[derive(Clone)]
pub struct EdgeRaw<Id: EdgeId, P> {
    to: u32,
    id: Id,
    payload: P,
}

impl<Id: EdgeId> EdgeRaw<Id, ()> {
    pub fn new(from: usize, to: usize) -> (usize, Self) {
        (
            from,
            Self {
                to: to as u32,
                id: Id::new(),
                payload: (),
            },
        )
    }
}

impl<Id: EdgeId, P> EdgeRaw<Id, P> {
    pub fn with_payload(from: usize, to: usize, payload: P) -> (usize, Self) {
        (from, Self::with_payload_impl(to, payload))
    }

    fn with_payload_impl(to: usize, payload: P) -> Self {
        Self {
            to: to as u32,
            id: Id::new(),
            payload,
        }
    }
}

impl<Id: EdgeId, P: Clone> EdgeTrait for EdgeRaw<Id, P> {
    type Payload = P;

    const REVERSABLE: bool = false;

    fn to(&self) -> usize {
        self.to as usize
    }

    fn id(&self) -> usize {
        self.id.id()
    }

    fn set_id(&mut self, id: usize) {
        self.id.set_id(id);
    }

    fn reverse_id(&self) -> usize {
        panic!("no reverse")
    }

    fn set_reverse_id(&mut self, _: usize) {
        panic!("no reverse")
    }

    fn reverse_edge(&self, _: usize) -> Self {
        panic!("no reverse")
    }

    fn payload(&self) -> &P {
        &self.payload
    }
}

pub type Edge<P> = EdgeRaw<NoId, P>;
pub type EdgeWithId<P> = EdgeRaw<WithId, P>;
}
pub mod edge_id {
pub trait EdgeId: Clone {
    fn new() -> Self;
    fn id(&self) -> usize;
    fn set_id(&mut self, id: usize);
}

#[derive(Clone)]
pub struct WithId {
    id: u32,
}

impl EdgeId for WithId {
    fn new() -> Self {
        Self { id: 0 }
    }

    fn id(&self) -> usize {
        self.id as usize
    }

    fn set_id(&mut self, id: usize) {
        self.id = id as u32;
    }
}

#[derive(Clone)]
pub struct NoId {}

impl EdgeId for NoId {
    fn new() -> Self {
        Self {}
    }

    fn id(&self) -> usize {
        panic!("Id called on no id")
    }

    fn set_id(&mut self, _: usize) {}
}
}
pub mod edge_trait {
pub trait EdgeTrait: Clone {
    type Payload;
    
    const REVERSABLE: bool;

    fn to(&self) -> usize;
    fn id(&self) -> usize;
    fn set_id(&mut self, id: usize);
    fn reverse_id(&self) -> usize;
    fn set_reverse_id(&mut self, reverse_id: usize);
    #[must_use]
    fn reverse_edge(&self, from: usize) -> Self;
    fn payload(&self) -> &Self::Payload;
}

pub trait BidirectionalEdgeTrait: EdgeTrait {}
}
}
}
pub mod io {
pub mod input {
use crate::algo_lib::collections::vec_ext::default::default_vec;
use std::io::Read;

pub struct Input<'s> {
    input: &'s mut (dyn Read + Send),
    buf: Vec<u8>,
    at: usize,
    buf_read: usize,
}

macro_rules! read_impl {
    ($t: ty, $read_name: ident, $read_vec_name: ident) => {
        pub fn $read_name(&mut self) -> $t {
            self.read()
        }

        pub fn $read_vec_name(&mut self, len: usize) -> Vec<$t> {
            self.read_vec(len)
        }
    };

    ($t: ty, $read_name: ident, $read_vec_name: ident, $read_pair_vec_name: ident) => {
        read_impl!($t, $read_name, $read_vec_name);

        pub fn $read_pair_vec_name(&mut self, len: usize) -> Vec<($t, $t)> {
            self.read_vec(len)
        }
    };
}

impl<'s> Input<'s> {
    const DEFAULT_BUF_SIZE: usize = 4096;

    pub fn new(input: &'s mut (dyn Read + Send)) -> Self {
        Self {
            input,
            buf: default_vec(Self::DEFAULT_BUF_SIZE),
            at: 0,
            buf_read: 0,
        }
    }

    pub fn new_with_size(input: &'s mut (dyn Read + Send), buf_size: usize) -> Self {
        Self {
            input,
            buf: default_vec(buf_size),
            at: 0,
            buf_read: 0,
        }
    }

    pub fn get(&mut self) -> Option<u8> {
        if self.refill_buffer() {
            let res = self.buf[self.at];
            self.at += 1;
            if res == b'\r' {
                if self.refill_buffer() && self.buf[self.at] == b'\n' {
                    self.at += 1;
                }
                return Some(b'\n');
            }
            Some(res)
        } else {
            None
        }
    }

    pub fn peek(&mut self) -> Option<u8> {
        if self.refill_buffer() {
            let res = self.buf[self.at];
            Some(if res == b'\r' { b'\n' } else { res })
        } else {
            None
        }
    }

    pub fn skip_whitespace(&mut self) {
        while let Some(b) = self.peek() {
            if !b.is_ascii_whitespace() {
                return;
            }
            self.get();
        }
    }

    pub fn next_token(&mut self) -> Option<Vec<u8>> {
        self.skip_whitespace();
        let mut res = Vec::new();
        while let Some(c) = self.get() {
            if c.is_ascii_whitespace() {
                break;
            }
            res.push(c);
        }
        if res.is_empty() {
            None
        } else {
            Some(res)
        }
    }

    //noinspection RsSelfConvention
    pub fn is_exhausted(&mut self) -> bool {
        self.peek().is_none()
    }

    //noinspection RsSelfConvention
    pub fn is_empty(&mut self) -> bool {
        self.skip_whitespace();
        self.is_exhausted()
    }

    pub fn read<T: Readable>(&mut self) -> T {
        T::read(self)
    }

    pub fn read_vec<T: Readable>(&mut self, size: usize) -> Vec<T> {
        let mut res = Vec::with_capacity(size);
        for _ in 0..size {
            res.push(self.read());
        }
        res
    }

    pub fn read_char(&mut self) -> u8 {
        self.skip_whitespace();
        self.get().unwrap()
    }

    read_impl!(u32, read_unsigned, read_unsigned_vec);
    read_impl!(u64, read_u64, read_u64_vec);
    read_impl!(usize, read_size, read_size_vec, read_size_pair_vec);
    read_impl!(i32, read_int, read_int_vec, read_int_pair_vec);
    read_impl!(i64, read_long, read_long_vec, read_long_pair_vec);
    read_impl!(i128, read_i128, read_i128_vec);

    fn refill_buffer(&mut self) -> bool {
        if self.at == self.buf_read {
            self.at = 0;
            self.buf_read = self.input.read(&mut self.buf).unwrap();
            self.buf_read != 0
        } else {
            true
        }
    }
}

pub trait Readable {
    fn read(input: &mut Input) -> Self;
}

impl Readable for u8 {
    fn read(input: &mut Input) -> Self {
        input.read_char()
    }
}

impl<T: Readable> Readable for Vec<T> {
    fn read(input: &mut Input) -> Self {
        let size = input.read();
        input.read_vec(size)
    }
}

macro_rules! read_integer {
    ($($t:ident)+) => {$(
        impl Readable for $t {
            fn read(input: &mut Input) -> Self {
                input.skip_whitespace();
                let mut c = input.get().unwrap();
                let sgn = match c {
                    b'-' => {
                        c = input.get().unwrap();
                        true
                    }
                    b'+' => {
                        c = input.get().unwrap();
                        false
                    }
                    _ => false,
                };
                let mut res = 0;
                loop {
                    assert!(c.is_ascii_digit());
                    res *= 10;
                    let d = (c - b'0') as $t;
                    if sgn {
                        res -= d;
                    } else {
                        res += d;
                    }
                    match input.get() {
                        None => break,
                        Some(ch) => {
                            if ch.is_ascii_whitespace() {
                                break;
                            } else {
                                c = ch;
                            }
                        }
                    }
                }
                res
            }
        }
    )+};
}

read_integer!(i8 i16 i32 i64 i128 isize u16 u32 u64 u128 usize);

macro_rules! tuple_readable {
    ($($name:ident)+) => {
        impl<$($name: Readable), +> Readable for ($($name,)+) {
            fn read(input: &mut Input) -> Self {
                ($($name::read(input),)+)
            }
        }
    }
}

tuple_readable! {T}
tuple_readable! {T U}
tuple_readable! {T U V}
tuple_readable! {T U V X}
tuple_readable! {T U V X Y}
tuple_readable! {T U V X Y Z}
tuple_readable! {T U V X Y Z A}
tuple_readable! {T U V X Y Z A B}
tuple_readable! {T U V X Y Z A B C}
tuple_readable! {T U V X Y Z A B C D}
tuple_readable! {T U V X Y Z A B C D E}
tuple_readable! {T U V X Y Z A B C D E F}

impl Read for Input<'_> {
    fn read(&mut self, buf: &mut [u8]) -> std::io::Result<usize> {
        if self.at == self.buf_read {
            self.input.read(buf)
        } else {
            let mut i = 0;
            while i < buf.len() && self.at < self.buf_read {
                buf[i] = self.buf[self.at];
                i += 1;
                self.at += 1;
            }
            Ok(i)
        }
    }
}
}
pub mod output {
use crate::algo_lib::collections::vec_ext::default::default_vec;
use std::cmp::Reverse;
use std::io::stderr;
use std::io::Stderr;
use std::io::Write;

#[derive(Copy, Clone)]
pub enum BoolOutput {
    YesNo,
    YesNoCaps,
    PossibleImpossible,
    Custom(&'static str, &'static str),
}

impl BoolOutput {
    pub fn output(&self, output: &mut Output, val: bool) {
        (if val { self.yes() } else { self.no() }).write(output);
    }

    fn yes(&self) -> &str {
        match self {
            BoolOutput::YesNo => "Yes",
            BoolOutput::YesNoCaps => "YES",
            BoolOutput::PossibleImpossible => "Possible",
            BoolOutput::Custom(yes, _) => yes,
        }
    }

    fn no(&self) -> &str {
        match self {
            BoolOutput::YesNo => "No",
            BoolOutput::YesNoCaps => "NO",
            BoolOutput::PossibleImpossible => "Impossible",
            BoolOutput::Custom(_, no) => no,
        }
    }
}

pub struct Output<'s> {
    output: &'s mut dyn Write,
    buf: Vec<u8>,
    at: usize,
    auto_flush: bool,
    bool_output: BoolOutput,
    precision: Option<usize>,
}

impl<'s> Output<'s> {
    const DEFAULT_BUF_SIZE: usize = 4096;

    pub fn new(output: &'s mut dyn Write) -> Self {
        Self {
            output,
            buf: default_vec(Self::DEFAULT_BUF_SIZE),
            at: 0,
            auto_flush: false,
            bool_output: BoolOutput::YesNoCaps,
            precision: None,
        }
    }

    pub fn new_with_auto_flush(output: &'s mut dyn Write) -> Self {
        Self {
            output,
            buf: default_vec(Self::DEFAULT_BUF_SIZE),
            at: 0,
            auto_flush: true,
            bool_output: BoolOutput::YesNoCaps,
            precision: None,
        }
    }

    pub fn flush(&mut self) {
        if self.at != 0 {
            self.output.write_all(&self.buf[..self.at]).unwrap();
            self.output.flush().unwrap();
            self.at = 0;
        }
    }

    pub fn print<T: Writable>(&mut self, s: T) {
        s.write(self);
        self.maybe_flush();
    }

    pub fn print_line<T: Writable>(&mut self, s: T) {
        self.print(s);
        self.put(b'\n');
        self.maybe_flush();
    }

    pub fn put(&mut self, b: u8) {
        self.buf[self.at] = b;
        self.at += 1;
        if self.at == self.buf.len() {
            self.flush();
        }
    }

    pub fn maybe_flush(&mut self) {
        if self.auto_flush {
            self.flush();
        }
    }

    pub fn print_per_line<T: Writable>(&mut self, arg: &[T]) {
        self.print_per_line_iter(arg.iter());
    }

    pub fn print_iter<T: Writable, I: Iterator<Item = T>>(&mut self, iter: I) {
        let mut first = true;
        for e in iter {
            if first {
                first = false;
            } else {
                self.put(b' ');
            }
            e.write(self);
        }
    }

    pub fn print_line_iter<T: Writable, I: Iterator<Item = T>>(&mut self, iter: I) {
        self.print_iter(iter);
        self.put(b'\n');
    }

    pub fn print_per_line_iter<T: Writable, I: Iterator<Item = T>>(&mut self, iter: I) {
        for e in iter {
            e.write(self);
            self.put(b'\n');
        }
    }

    pub fn set_bool_output(&mut self, bool_output: BoolOutput) {
        self.bool_output = bool_output;
    }
    pub fn set_precision(&mut self, precision: Option<usize>) {
        self.precision = precision;
    }
    pub fn get_precision(&self) -> Option<usize> {
        self.precision
    }
}

impl Write for Output<'_> {
    fn write(&mut self, buf: &[u8]) -> std::io::Result<usize> {
        let mut start = 0usize;
        let mut rem = buf.len();
        while rem > 0 {
            let len = (self.buf.len() - self.at).min(rem);
            self.buf[self.at..self.at + len].copy_from_slice(&buf[start..start + len]);
            self.at += len;
            if self.at == self.buf.len() {
                self.flush();
            }
            start += len;
            rem -= len;
        }
        self.maybe_flush();
        Ok(buf.len())
    }

    fn flush(&mut self) -> std::io::Result<()> {
        self.flush();
        Ok(())
    }
}

pub trait Writable {
    fn write(&self, output: &mut Output);
}

impl Writable for &str {
    fn write(&self, output: &mut Output) {
        output.write_all(self.as_bytes()).unwrap();
    }
}

impl Writable for String {
    fn write(&self, output: &mut Output) {
        output.write_all(self.as_bytes()).unwrap();
    }
}

impl Writable for char {
    fn write(&self, output: &mut Output) {
        output.put(*self as u8);
    }
}

impl Writable for u8 {
    fn write(&self, output: &mut Output) {
        output.put(*self);
    }
}

impl<T: Writable> Writable for [T] {
    fn write(&self, output: &mut Output) {
        output.print_iter(self.iter());
    }
}

impl<T: Writable, const N: usize> Writable for [T; N] {
    fn write(&self, output: &mut Output) {
        output.print_iter(self.iter());
    }
}

impl<T: Writable + ?Sized> Writable for &T {
    fn write(&self, output: &mut Output) {
        T::write(self, output)
    }
}

impl<T: Writable> Writable for Vec<T> {
    fn write(&self, output: &mut Output) {
        self.as_slice().write(output);
    }
}

impl Writable for () {
    fn write(&self, _output: &mut Output) {}
}

macro_rules! write_to_string {
    ($($t:ident)+) => {$(
        impl Writable for $t {
            fn write(&self, output: &mut Output) {
                self.to_string().write(output);
            }
        }
    )+};
}

write_to_string!(u16 u32 u64 u128 usize i8 i16 i32 i64 i128 isize);

macro_rules! tuple_writable {
    ($name0:ident $($name:ident: $id:tt )*) => {
        impl<$name0: Writable, $($name: Writable,)*> Writable for ($name0, $($name,)*) {
            fn write(&self, out: &mut Output) {
                self.0.write(out);
                $(
                out.put(b' ');
                self.$id.write(out);
                )*
            }
        }
    }
}

tuple_writable! {T}
tuple_writable! {T U:1}
tuple_writable! {T U:1 V:2}
tuple_writable! {T U:1 V:2 X:3}
tuple_writable! {T U:1 V:2 X:3 Y:4}
tuple_writable! {T U:1 V:2 X:3 Y:4 Z:5}
tuple_writable! {T U:1 V:2 X:3 Y:4 Z:5 A:6}
tuple_writable! {T U:1 V:2 X:3 Y:4 Z:5 A:6 B:7}
tuple_writable! {T U:1 V:2 X:3 Y:4 Z:5 A:6 B:7 C:8}

impl<T: Writable> Writable for Option<T> {
    fn write(&self, output: &mut Output) {
        match self {
            None => (-1).write(output),
            Some(t) => t.write(output),
        }
    }
}

impl Writable for bool {
    fn write(&self, output: &mut Output) {
        let bool_output = output.bool_output;
        bool_output.output(output, *self)
    }
}

impl<T: Writable> Writable for Reverse<T> {
    fn write(&self, output: &mut Output) {
        self.0.write(output);
    }
}

static mut ERR: Option<Stderr> = None;

pub fn err() -> Output<'static> {
    unsafe {
        if ERR.is_none() {
            ERR = Some(stderr());
        }
        Output::new_with_auto_flush(ERR.as_mut().unwrap())
    }
}
}
}
pub mod misc {
pub mod recursive_function {
use std::marker::PhantomData;

macro_rules! recursive_function {
    ($name: ident, $trait: ident, ($($type: ident $arg: ident,)*)) => {
        pub trait $trait<$($type, )*Output> {
            fn call(&mut self, $($arg: $type,)*) -> Output;
        }

        pub struct $name<F, $($type, )*Output>
        where
            F: FnMut(&mut dyn $trait<$($type, )*Output>, $($type, )*) -> Output,
        {
            f: std::cell::UnsafeCell<F>,
            $($arg: PhantomData<$type>,
            )*
            phantom_output: PhantomData<Output>,
        }

        impl<F, $($type, )*Output> $name<F, $($type, )*Output>
        where
            F: FnMut(&mut dyn $trait<$($type, )*Output>, $($type, )*) -> Output,
        {
            pub fn new(f: F) -> Self {
                Self {
                    f: std::cell::UnsafeCell::new(f),
                    $($arg: Default::default(),
                    )*
                    phantom_output: Default::default(),
                }
            }
        }

        impl<F, $($type, )*Output> $trait<$($type, )*Output> for $name<F, $($type, )*Output>
        where
            F: FnMut(&mut dyn $trait<$($type, )*Output>, $($type, )*) -> Output,
        {
            fn call(&mut self, $($arg: $type,)*) -> Output {
                unsafe { (*self.f.get())(self, $($arg, )*) }
            }
        }
    }
}

recursive_function!(RecursiveFunction0, Callable0, ());
recursive_function!(RecursiveFunction, Callable, (Arg arg,));
recursive_function!(RecursiveFunction2, Callable2, (Arg1 arg1, Arg2 arg2,));
recursive_function!(RecursiveFunction3, Callable3, (Arg1 arg1, Arg2 arg2, Arg3 arg3,));
recursive_function!(RecursiveFunction4, Callable4, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4,));
recursive_function!(RecursiveFunction5, Callable5, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4, Arg5 arg5,));
recursive_function!(RecursiveFunction6, Callable6, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4, Arg5 arg5, Arg6 arg6,));
recursive_function!(RecursiveFunction7, Callable7, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4, Arg5 arg5, Arg6 arg6, Arg7 arg7,));
recursive_function!(RecursiveFunction8, Callable8, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4, Arg5 arg5, Arg6 arg6, Arg7 arg7, Arg8 arg8,));
recursive_function!(RecursiveFunction9, Callable9, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4, Arg5 arg5, Arg6 arg6, Arg7 arg7, Arg8 arg8, Arg9 arg9,));
}
pub mod test_type {
pub enum TestType {
    Single,
    MultiNumber,
    MultiEof,
}

pub enum TaskType {
    Classic,
    Interactive,
}
}
}
pub mod numbers {
pub mod num_traits {
pub mod algebra {
use crate::algo_lib::numbers::num_traits::invertible::Invertible;
use std::ops::Add;
use std::ops::AddAssign;
use std::ops::Div;
use std::ops::DivAssign;
use std::ops::Mul;
use std::ops::MulAssign;
use std::ops::Neg;
use std::ops::Rem;
use std::ops::RemAssign;
use std::ops::Sub;
use std::ops::SubAssign;

pub trait Zero {
    fn zero() -> Self;
}

pub trait One {
    fn one() -> Self;
}

pub trait AdditionMonoid: Add<Output = Self> + AddAssign + Zero + Eq + Sized {}

impl<T: Add<Output = Self> + AddAssign + Zero + Eq> AdditionMonoid for T {}

pub trait AdditionMonoidWithSub: AdditionMonoid + Sub<Output = Self> + SubAssign {}

impl<T: AdditionMonoid + Sub<Output = Self> + SubAssign> AdditionMonoidWithSub for T {}

pub trait AdditionGroup: AdditionMonoidWithSub + Neg<Output = Self> {}

impl<T: AdditionMonoidWithSub + Neg<Output = Self>> AdditionGroup for T {}

pub trait MultiplicationMonoid: Mul<Output = Self> + MulAssign + One + Eq + Sized {}

impl<T: Mul<Output = Self> + MulAssign + One + Eq> MultiplicationMonoid for T {}

pub trait IntegerMultiplicationMonoid:
    MultiplicationMonoid + Div<Output = Self> + Rem<Output = Self> + DivAssign + RemAssign
{
}

impl<T: MultiplicationMonoid + Div<Output = Self> + Rem<Output = Self> + DivAssign + RemAssign>
    IntegerMultiplicationMonoid for T
{
}

pub trait MultiplicationGroup:
    MultiplicationMonoid + Div<Output = Self> + DivAssign + Invertible<Output = Self>
{
}

impl<T: MultiplicationMonoid + Div<Output = Self> + DivAssign + Invertible<Output = Self>>
    MultiplicationGroup for T
{
}

pub trait SemiRing: AdditionMonoid + MultiplicationMonoid {}

impl<T: AdditionMonoid + MultiplicationMonoid> SemiRing for T {}

pub trait SemiRingWithSub: AdditionMonoidWithSub + SemiRing {}

impl<T: AdditionMonoidWithSub + SemiRing> SemiRingWithSub for T {}

pub trait Ring: SemiRing + AdditionGroup {}

impl<T: SemiRing + AdditionGroup> Ring for T {}

pub trait IntegerSemiRing: SemiRing + IntegerMultiplicationMonoid {}

impl<T: SemiRing + IntegerMultiplicationMonoid> IntegerSemiRing for T {}

pub trait IntegerSemiRingWithSub: SemiRingWithSub + IntegerSemiRing {}

impl<T: SemiRingWithSub + IntegerSemiRing> IntegerSemiRingWithSub for T {}

pub trait IntegerRing: IntegerSemiRing + Ring {}

impl<T: IntegerSemiRing + Ring> IntegerRing for T {}

pub trait Field: Ring + MultiplicationGroup {}

impl<T: Ring + MultiplicationGroup> Field for T {}

macro_rules! zero_one_integer_impl {
    ($($t: ident)+) => {$(
        impl Zero for $t {
            fn zero() -> Self {
                0
            }
        }

        impl One for $t {
            fn one() -> Self {
                1
            }
        }
    )+};
}

zero_one_integer_impl!(i128 i64 i32 i16 i8 isize u128 u64 u32 u16 u8 usize);
}
pub mod bit_ops {
use crate::algo_lib::numbers::num_traits::algebra::One;
use crate::algo_lib::numbers::num_traits::algebra::Zero;
use std::ops::BitAnd;
use std::ops::BitAndAssign;
use std::ops::BitOr;
use std::ops::BitOrAssign;
use std::ops::BitXor;
use std::ops::BitXorAssign;
use std::ops::Not;
use std::ops::RangeInclusive;
use std::ops::Shl;
use std::ops::Sub;
use std::ops::ShlAssign;
use std::ops::Shr;
use std::ops::ShrAssign;

pub trait BitOps:
    Copy
    + BitAnd<Output = Self>
    + BitAndAssign
    + BitOr<Output = Self>
    + BitOrAssign
    + BitXor<Output = Self>
    + BitXorAssign
    + Not<Output = Self>
    + Shl<usize, Output = Self>
    + ShlAssign<usize>
    + Shr<usize, Output = Self>
    + ShrAssign<usize>
    + Zero
    + One
    + PartialEq
{
    fn bit(at: usize) -> Self {
        Self::one() << at
    }

    fn is_set(&self, at: usize) -> bool {
        (*self >> at & Self::one()) == Self::one()
    }

    fn set_bit(&mut self, at: usize) {
        *self |= Self::bit(at)
    }

    fn unset_bit(&mut self, at: usize) {
        *self &= !Self::bit(at)
    }

    #[must_use]
    fn with_bit(mut self, at: usize) -> Self {
        self.set_bit(at);
        self
    }

    #[must_use]
    fn without_bit(mut self, at: usize) -> Self {
        self.unset_bit(at);
        self
    }

    fn flip_bit(&mut self, at: usize) {
        *self ^= Self::bit(at)
    }

    #[must_use]
    fn flipped_bit(mut self, at: usize) -> Self {
        self.flip_bit(at);
        self
    }

    fn all_bits(n: usize) -> Self {
        let mut res = Self::zero();
        for i in 0..n {
            res.set_bit(i);
        }
        res
    }

    fn iter_all(n: usize) -> RangeInclusive<Self> {
        Self::zero()..=Self::all_bits(n)
    }
}

pub struct BitIter<T> {
    cur: T,
    all: T,
    ended: bool,
}

impl<T: Copy> BitIter<T> {
    pub fn new(all: T) -> Self {
        Self {
            cur: all,
            all,
            ended: false,
        }
    }
}

impl<T: BitOps + Sub<Output = T>> Iterator for BitIter<T> {
    type Item = T;

    fn next(&mut self) -> Option<Self::Item> {
        if self.ended {
            return None;
        }
        let res = self.cur;
        if self.cur == T::zero() {
            self.ended = true;
        } else {
            self.cur = (self.cur - T::one()) & self.all;
        }
        Some(res)
    }
}

impl<
        T: Copy
            + BitAnd<Output = Self>
            + BitAndAssign
            + BitOr<Output = Self>
            + BitOrAssign
            + BitXor<Output = Self>
            + BitXorAssign
            + Not<Output = Self>
            + Shl<usize, Output = Self>
            + ShlAssign<usize>
            + Shr<usize, Output = Self>
            + ShrAssign<usize>
            + One
            + Zero
            + PartialEq,
    > BitOps for T
{
}

pub trait Bits: BitOps {
    fn bits() -> u32;
}

macro_rules! bits_integer_impl {
    ($($t: ident $bits: expr),+) => {$(
        impl Bits for $t {
            fn bits() -> u32 {
                $bits
            }
        }
    )+};
}

bits_integer_impl!(i128 128, i64 64, i32 32, i16 16, i8 8, isize 64, u128 128, u64 64, u32 32, u16 16, u8 8, usize 64);
}
pub mod invertible {
pub trait Invertible {
    type Output;

    fn inv(&self) -> Option<Self::Output>;
}
}
}
}
pub mod string {
pub mod str {
use crate::algo_lib::collections::iter_ext::collect::IterCollect;
use crate::algo_lib::collections::slice_ext::backward::Back;
use crate::algo_lib::io::input::Input;
use crate::algo_lib::io::input::Readable;
use crate::algo_lib::io::output::Output;
use crate::algo_lib::io::output::Writable;
use std::cmp::Ordering;
use std::fmt::Debug;
use std::fmt::Display;
use std::fmt::Formatter;
use std::hash::Hash;
use std::hash::Hasher;
use std::iter::Copied;
use std::iter::FromIterator;
use std::marker::PhantomData;
use std::ops::Add;
use std::ops::AddAssign;
use std::ops::Deref;
use std::ops::DerefMut;
use std::ops::Index;
use std::ops::IndexMut;
use std::ops::RangeBounds;
use std::slice::Iter;
use std::slice::IterMut;
use std::slice::SliceIndex;
use std::str::FromStr;
use std::vec::IntoIter;

pub enum Str<'s> {
    Extendable(Vec<u8>, PhantomData<&'s [u8]>),
    Owned(Box<[u8]>, PhantomData<&'s [u8]>),
    Ref(&'s [u8]),
}

impl<'s> Str<'s> {
    pub fn substr(&self, range: impl RangeBounds<usize>) -> Str {
        let from = match range.start_bound() {
            std::ops::Bound::Included(&i) => i,
            std::ops::Bound::Excluded(&i) => i + 1,
            std::ops::Bound::Unbounded => 0,
        };
        let to = match range.end_bound() {
            std::ops::Bound::Included(&i) => i + 1,
            std::ops::Bound::Excluded(&i) => i,
            std::ops::Bound::Unbounded => self.len(),
        };
        Str::from(&self[from..to])
    }
}

impl Default for Str<'static> {
    fn default() -> Self {
        Self::new()
    }
}

impl Str<'static> {
    pub fn new() -> Self {
        Str::Extendable(Vec::new(), PhantomData)
    }

    pub fn with_capacity(cap: usize) -> Self {
        Str::Extendable(Vec::with_capacity(cap), PhantomData)
    }
}

impl<'s> Str<'s> {
    pub fn push(&mut self, c: u8) {
        self.transform_to_extendable();
        self.as_extendable().push(c)
    }

    pub fn pop(&mut self) -> Option<u8> {
        self.transform_to_extendable();
        self.as_extendable().pop()
    }

    pub fn as_slice(&self) -> &[u8] {
        match self {
            Str::Extendable(s, _) => s.as_ref(),
            Str::Owned(s, _) => s.as_ref(),
            Str::Ref(s) => s,
        }
    }

    pub fn len(&self) -> usize {
        self.as_slice().len()
    }

    pub fn is_empty(&self) -> bool {
        self.len() == 0
    }

    pub fn resize(&mut self, new_len: usize, value: u8) {
        self.transform_to_extendable();
        self.as_extendable().resize(new_len, value);
    }

    pub fn iter(&self) -> Copied<Iter<u8>> {
        match self {
            Str::Extendable(v, _) => v.iter(),
            Str::Owned(v, _) => v.iter(),
            Str::Ref(v) => v.iter(),
        }
        .copied()
    }

    pub fn iter_mut(&mut self) -> IterMut<u8> {
        self.transform_to_owned();
        self.as_mut_slice().iter_mut()
    }

    pub fn sort(&mut self) {
        self.transform_to_owned();
        self.as_mut_slice().sort_unstable();
    }

    pub fn into_owned(mut self) -> Str<'static> {
        self.transform_to_owned();
        match self {
            Str::Extendable(v, _) => Str::Extendable(v, PhantomData),
            Str::Owned(v, _) => Str::Owned(v, PhantomData),
            _ => unreachable!(),
        }
    }

    fn transform_to_extendable(&mut self) {
        match self {
            Str::Extendable(_, _) => {}
            Str::Owned(_, _) => {
                let mut fake = Str::new();
                std::mem::swap(self, &mut fake);
                if let Str::Owned(s, _) = fake {
                    *self = Str::Extendable(s.into_vec(), PhantomData)
                } else {
                    unreachable!();
                }
            }
            Str::Ref(s) => *self = Str::Extendable(s.to_vec(), PhantomData),
        }
    }

    fn as_extendable(&mut self) -> &mut Vec<u8> {
        match self {
            Str::Extendable(s, _) => s,
            _ => panic!("unreachable"),
        }
    }

    fn transform_to_owned(&mut self) {
        if let Str::Ref(s) = self {
            *self = Str::Owned(s.to_vec().into_boxed_slice(), PhantomData)
        }
    }

    pub fn as_mut_slice(&mut self) -> &mut [u8] {
        self.transform_to_owned();
        match self {
            Str::Extendable(s, _) => s.as_mut_slice(),
            Str::Owned(s, _) => s.as_mut(),
            _ => panic!("unreachable"),
        }
    }

    pub fn into_string(self) -> String {
        match self {
            Str::Extendable(v, _) => unsafe { String::from_utf8_unchecked(v) },
            Str::Owned(v, _) => unsafe { String::from_utf8_unchecked(v.into_vec()) },
            Str::Ref(v) => String::from_utf8_lossy(v).into_owned(),
        }
    }

    pub fn reverse(&mut self) {
        self.as_mut_slice().reverse();
    }

    pub fn trim(&self) -> Str<'_> {
        let mut start = 0;
        let mut end = self.len();
        while start < end && (self[start] as char).is_whitespace() {
            start += 1;
        }
        while start < end && (self[end - 1] as char).is_whitespace() {
            end -= 1;
        }
        self[start..end].into()
    }

    pub fn split<'a, 'b>(&'a self, sep: impl Into<Str<'b>>) -> Vec<Str<'a>>
    where
        's: 'a,
    {
        let sep = sep.into();
        let mut res = Vec::new();
        let mut start = 0;
        for i in 0..self.len() {
            if self[i..].starts_with(sep.as_slice()) {
                res.push(self[start..i].into());
                start = i + sep.len();
            }
        }
        res.push(self[start..].into());
        res
    }

    pub fn parse<F: FromStr>(self) -> F
    where
        F::Err: Debug,
    {
        self.into_string().parse().unwrap()
    }

    pub fn parse_vec<T: Readable>(&self) -> Vec<T> {
        let mut bytes = self.as_slice();
        let mut input = Input::new(&mut bytes);
        let mut res = Vec::new();
        while !input.is_exhausted() {
            res.push(input.read());
        }
        res
    }

    pub fn qty(&self, from: u8, to: u8) -> Vec<usize> {
        let mut res = vec![0; (to - from + 1) as usize];
        for &c in self.as_slice() {
            res[(c - from) as usize] += 1;
        }
        res
    }

    pub fn qty_lower(&self) -> Vec<usize> {
        self.qty(b'a', b'z')
    }
}

impl<'s> IntoIterator for Str<'s> {
    type Item = u8;
    type IntoIter = IntoIter<u8>;

    #[allow(clippy::unnecessary_to_owned)]
    fn into_iter(self) -> Self::IntoIter {
        match self {
            Str::Extendable(v, _) => v.into_iter(),
            Str::Owned(v, _) => v.into_vec().into_iter(),
            Str::Ref(v) => v.to_vec().into_iter(),
        }
    }
}

impl From<String> for Str<'static> {
    fn from(s: String) -> Self {
        Str::Extendable(s.into(), PhantomData)
    }
}

impl<'s> From<&'s str> for Str<'s> {
    fn from(s: &'s str) -> Self {
        Str::Ref(s.as_bytes())
    }
}

impl From<Vec<u8>> for Str<'static> {
    fn from(s: Vec<u8>) -> Self {
        Str::Extendable(s, PhantomData)
    }
}

impl<'s> From<&'s [u8]> for Str<'s> {
    fn from(s: &'s [u8]) -> Self {
        Str::Ref(s)
    }
}

impl<'s, const N: usize> From<&'s [u8; N]> for Str<'s> {
    fn from(s: &'s [u8; N]) -> Self {
        Str::Ref(s)
    }
}

impl<'s> From<&'s String> for Str<'s> {
    fn from(s: &'s String) -> Self {
        Str::Ref(s.as_bytes())
    }
}

impl<'s> From<&'s Vec<u8>> for Str<'s> {
    fn from(s: &'s Vec<u8>) -> Self {
        Str::Ref(s.as_slice())
    }
}

impl From<u8> for Str<'static> {
    fn from(c: u8) -> Self {
        Str::Owned(Box::new([c]), PhantomData)
    }
}

impl From<char> for Str<'static> {
    fn from(c: char) -> Self {
        Str::from(c as u8)
    }
}

impl<'s, 't: 's> From<&'s Str<'t>> for Str<'s> {
    fn from(value: &'s Str<'t>) -> Self {
        Str::Ref(value.as_slice())
    }
}

impl<R: SliceIndex<[u8]>> Index<R> for Str<'_> {
    type Output = R::Output;

    fn index(&self, index: R) -> &Self::Output {
        self.as_slice().index(index)
    }
}

impl<R: SliceIndex<[u8]>> IndexMut<R> for Str<'_> {
    fn index_mut(&mut self, index: R) -> &mut Self::Output {
        self.transform_to_owned();
        self.as_mut_slice().index_mut(index)
    }
}

impl Clone for Str<'_> {
    fn clone(&self) -> Self {
        match self {
            Str::Extendable(s, _) => s.clone().into(),
            Str::Owned(s, _) => s.to_vec().into(),
            Str::Ref(s) => Str::Ref(s),
        }
    }
}

impl<'r, 's, S: Into<Str<'r>>> AddAssign<S> for Str<'s> {
    fn add_assign(&mut self, rhs: S) {
        self.transform_to_extendable();
        self.as_extendable()
            .extend_from_slice(rhs.into().as_slice());
    }
}

impl<'r, 's, S: Into<Str<'r>>> Add<S> for Str<'s> {
    type Output = Str<'s>;

    fn add(mut self, rhs: S) -> Self::Output {
        self += rhs;
        self
    }
}

impl Readable for Str<'static> {
    fn read(input: &mut Input) -> Self {
        input.next_token().unwrap().into()
    }
}

impl Writable for Str<'_> {
    fn write(&self, output: &mut Output) {
        for c in self.as_slice() {
            output.put(*c);
        }
        output.maybe_flush();
    }
}

impl Display for Str<'_> {
    fn fmt(&self, f: &mut Formatter<'_>) -> std::fmt::Result {
        <String as Display>::fmt(&String::from_utf8(self.as_slice().to_vec()).unwrap(), f)
    }
}

impl Hash for Str<'_> {
    fn hash<H: Hasher>(&self, state: &mut H) {
        self.as_slice().hash(state);
    }
}

impl<'r> PartialEq<Str<'r>> for Str<'_> {
    fn eq(&self, other: &Str<'r>) -> bool {
        self.as_slice().eq(other.as_slice())
    }
}

impl Eq for Str<'_> {}

impl<'r> PartialOrd<Str<'r>> for Str<'_> {
    fn partial_cmp(&self, other: &Str<'r>) -> Option<Ordering> {
        self.as_slice().partial_cmp(other.as_slice())
    }
}

impl Ord for Str<'_> {
    fn cmp(&self, other: &Self) -> Ordering {
        self.as_slice().cmp(other.as_slice())
    }
}

impl FromIterator<u8> for Str<'static> {
    fn from_iter<T: IntoIterator<Item = u8>>(iter: T) -> Self {
        Self::Extendable(iter.into_iter().collect_vec(), Default::default())
    }
}

impl<'r> FromIterator<&'r u8> for Str<'static> {
    fn from_iter<T: IntoIterator<Item = &'r u8>>(iter: T) -> Self {
        Self::Extendable(iter.into_iter().cloned().collect_vec(), Default::default())
    }
}

impl Deref for Str<'_> {
    type Target = [u8];

    fn deref(&self) -> &Self::Target {
        self.as_slice()
    }
}

impl DerefMut for Str<'_> {
    fn deref_mut(&mut self) -> &mut Self::Target {
        self.as_mut_slice()
    }
}

pub trait StrReader {
    fn read_str(&mut self) -> Str<'static>;
    fn read_str_vec(&mut self, n: usize) -> Vec<Str<'static>>;
    fn read_line(&mut self) -> Str<'static>;
    fn read_line_vec(&mut self, n: usize) -> Vec<Str<'static>>;
    fn read_lines(&mut self) -> Vec<Str<'static>>;
}

impl StrReader for Input<'_> {
    fn read_str(&mut self) -> Str<'static> {
        self.read()
    }

    fn read_str_vec(&mut self, n: usize) -> Vec<Str<'static>> {
        self.read_vec(n)
    }

    fn read_line(&mut self) -> Str<'static> {
        let mut res = Str::new();
        while let Some(c) = self.get() {
            if c == b'\n' {
                break;
            }
            res.push(c);
        }
        res
    }

    fn read_line_vec(&mut self, n: usize) -> Vec<Str<'static>> {
        let mut res = Vec::with_capacity(n);
        for _ in 0..n {
            res.push(self.read_line());
        }
        res
    }

    fn read_lines(&mut self) -> Vec<Str<'static>> {
        let mut res = Vec::new();
        while !self.is_exhausted() {
            res.push(self.read_line());
        }
        if let Some(s) = res.last() {
            if s.is_empty() {
                res.pop();
            }
        }
        res
    }
}

impl Index<Back> for Str<'_> {
    type Output = u8;

    fn index(&self, index: Back) -> &Self::Output {
        &self[self.len() - index.0 - 1]
    }
}

impl IndexMut<Back> for Str<'_> {
    fn index_mut(&mut self, index: Back) -> &mut Self::Output {
        let len = self.len();
        &mut self[len - index.0 - 1]
    }
}

impl AsRef<[u8]> for Str<'_> {
    fn as_ref(&self) -> &[u8] {
        self.as_slice()
    }
}
}
}
}
fn main() {
    let mut sin = std::io::stdin();
    let input = algo_lib::io::input::Input::new(&mut sin);
    let mut stdout = std::io::stdout();
    let output = algo_lib::io::output::Output::new(&mut stdout);
    solution::run(input, output);
}