// 
pub mod solution {
//{"name":"ucup_10_g","group":"Manual","url":"","interactive":false,"timeLimit":2000,"tests":[{"input":"","output":""}],"testType":"multiNumber","input":{"type":"stdin","fileName":null,"pattern":null},"output":{"type":"stdout","fileName":null,"pattern":null},"languages":{"java":{"taskClass":"ucup_10_g"}}}

use crate::algo_lib::collections::bit_set::BitSet;
use crate::algo_lib::collections::iter_ext::collect::IterCollect;
use crate::algo_lib::collections::vec_ext::inc_dec::IncDec;
use crate::algo_lib::graph::edges::edge_trait::EdgeTrait;
use crate::algo_lib::graph::edges::flow_edge::FlowEdge;
use crate::algo_lib::graph::edges::flow_edge_trait::FlowEdgeTrait;
use crate::algo_lib::graph::graph::Graph;
use crate::algo_lib::graph::max_flow::MaxFlow;
use crate::algo_lib::io::input::Input;
use crate::algo_lib::io::output::Output;
use crate::algo_lib::misc::test_type::TaskType;
use crate::algo_lib::misc::test_type::TestType;

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 castles = input.read_size_pair_vec(n);
    let obstacles = input.read_size_pair_vec(m);

    let mut base = 0;
    #[derive(Copy, Clone)]
    enum ObjectType {
        Castle,
        Obstacle,
    }
    #[derive(Copy, Clone)]
    struct Object {
        id: usize,
        x: usize,
        y: usize,
        t: ObjectType,
    }

    let mut objects = castles
        .iter()
        .enumerate()
        .map(|(i, &(x, y))| Object {
            id: i,
            x,
            y,
            t: ObjectType::Castle,
        })
        .chain(obstacles.iter().enumerate().map(|(i, &(x, y))| Object {
            id: i,
            x,
            y,
            t: ObjectType::Obstacle,
        }))
        .collect::<Vec<_>>();
    objects.sort_by_key(|o| (o.x, o.y));
    let mut last_x = 0;
    let mut obstacles_in_gap = Vec::new();
    let mut castle_started = false;
    let mut hor_gap_id = 0;
    let mut obst_hor_gap = vec![None; m];
    for obj in &objects {
        if obj.x != last_x {
            obstacles_in_gap.clear();
            castle_started = false;
        }
        match obj.t {
            ObjectType::Castle => {
                if castle_started {
                    if obstacles_in_gap.is_empty() {
                        base += 1;
                    } else {
                        for &obstacle in &obstacles_in_gap {
                            obst_hor_gap[obstacle] = Some(hor_gap_id);
                        }
                        obstacles_in_gap.clear();
                        hor_gap_id += 1;
                    }
                }
                castle_started = true;
            }
            ObjectType::Obstacle => {
                if castle_started {
                    obstacles_in_gap.push(obj.id);
                }
            }
        }
        last_x = obj.x;
    }
    objects.sort_by_key(|o| (o.y, o.x));
    let mut last_y = 0;
    let mut obstacles_in_gap = Vec::new();
    let mut castle_started = false;
    let mut ver_gap_id = 0;
    let mut obst_ver_gap = vec![None; m];
    for obj in &objects {
        if obj.y != last_y {
            obstacles_in_gap.clear();
            castle_started = false;
        }
        match obj.t {
            ObjectType::Castle => {
                if castle_started {
                    if obstacles_in_gap.is_empty() {
                        base += 1;
                    } else {
                        for &obstacle in &obstacles_in_gap {
                            obst_ver_gap[obstacle] = Some(ver_gap_id);
                        }
                        obstacles_in_gap.clear();
                        ver_gap_id += 1;
                    }
                }
                castle_started = true;
            }
            ObjectType::Obstacle => {
                if castle_started {
                    obstacles_in_gap.push(obj.id);
                }
            }
        }
        last_y = obj.y;
    }
    let mut graph = Graph::new(hor_gap_id + ver_gap_id + 3);
    let source = hor_gap_id + ver_gap_id;
    let sink = source + 1;
    let real_source = sink + 1;
    for i in 0..hor_gap_id {
        graph.add_edge(FlowEdge::with_payload(source, i, 1, 0));
    }
    for i in 0..ver_gap_id {
        graph.add_edge(FlowEdge::with_payload(hor_gap_id + i, sink, 1, 0));
    }
    for i in 0..m {
        if let Some(hor_gap) = obst_hor_gap[i] {
            if let Some(ver_gap) = obst_ver_gap[i] {
                graph.add_edge(FlowEdge::with_payload(hor_gap, hor_gap_id + ver_gap, 1, i));
            }
        }
    }
    graph.add_edge(FlowEdge::with_payload(real_source, source, m - k, 0));
    let mut ans = base + hor_gap_id + ver_gap_id;
    let max_flow = graph.max_flow(real_source, sink);
    ans -= 2 * max_flow;
    let mut to_remove = BitSet::new(m);
    to_remove.fill(true);
    let mut left_done = BitSet::new(hor_gap_id);
    let mut right_done = BitSet::new(ver_gap_id);
    for i in 0..hor_gap_id {
        for e in &graph[i] {
            if e.to() != source && e.flow(&graph) == 1 {
                to_remove.unset(*e.payload());
                left_done.set(i);
                right_done.set(e.to() - hor_gap_id);
            }
        }
    }
    let mut can_do = m - k - max_flow;
    for i in 0..m {
        if let Some(hor_gap) = obst_hor_gap[i] {
            if !left_done[hor_gap] && can_do > 0 {
                assert!(to_remove[i]);
                can_do -= 1;
                to_remove.unset(i);
                left_done.set(hor_gap);
                ans -= 1;
            }
        }
        if let Some(ver_gap) = obst_ver_gap[i] {
            if !right_done[ver_gap] && can_do > 0 {
                assert!(to_remove[i]);
                can_do -= 1;
                to_remove.unset(i);
                right_done.set(ver_gap);
                ans -= 1;
            }
        }
    }
    for i in 0..m {
        if to_remove[i] && can_do > 0 {
            can_do -= 1;
            to_remove.unset(i);
        }
    }
    out.print_line(ans);
    out.print_line(to_remove.iter().collect_vec().inc());
}

pub static TEST_TYPE: TestType = TestType::MultiNumber;
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.skip_whitespace();
            input.peek().is_none()
        }
        TaskType::Interactive => true,
    }
}

}
pub mod algo_lib {
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 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 slice_ext {
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 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 inc_dec {
use crate::algo_lib::numbers::num_traits::algebra::AdditionMonoidWithSub;
use crate::algo_lib::numbers::num_traits::algebra::One;

pub trait IncDec {
    #[must_use]
    fn inc(self) -> Self;
    #[must_use]
    fn dec(self) -> Self;
}

impl<T: AdditionMonoidWithSub + One> IncDec for T {
    fn inc(self) -> Self {
        self + T::one()
    }

    fn dec(self) -> Self {
        self - T::one()
    }
}

impl<T: AdditionMonoidWithSub + One> IncDec for Vec<T> {
    fn inc(mut self) -> Self {
        self.iter_mut().for_each(|i| *i += T::one());
        self
    }

    fn dec(mut self) -> Self {
        self.iter_mut().for_each(|i| *i -= T::one());
        self
    }
}

impl<T: AdditionMonoidWithSub + One, U: AdditionMonoidWithSub + One> IncDec for Vec<(T, U)> {
    fn inc(mut self) -> Self {
        self.iter_mut().for_each(|(i, j)| {
            *i += T::one();
            *j += U::one();
        });
        self
    }

    fn dec(mut self) -> Self {
        self.iter_mut().for_each(|(i, j)| {
            *i -= T::one();
            *j -= U::one();
        });
        self
    }
}

impl<T: AdditionMonoidWithSub + One, U: AdditionMonoidWithSub + One, V> IncDec for Vec<(T, U, V)> {
    fn inc(mut self) -> Self {
        self.iter_mut().for_each(|(i, j, _)| {
            *i += T::one();
            *j += U::one();
        });
        self
    }

    fn dec(mut self) -> Self {
        self.iter_mut().for_each(|(i, j, _)| {
            *i -= T::one();
            *j -= U::one();
        });
        self
    }
}

impl<T: AdditionMonoidWithSub + One, U: AdditionMonoidWithSub + One, V, W> IncDec
    for Vec<(T, U, V, W)>
{
    fn inc(mut self) -> Self {
        self.iter_mut().for_each(|(i, j, ..)| {
            *i += T::one();
            *j += U::one();
        });
        self
    }

    fn dec(mut self) -> Self {
        self.iter_mut().for_each(|(i, j, ..)| {
            *i -= T::one();
            *j -= U::one();
        });
        self
    }
}

impl<T: AdditionMonoidWithSub + One, U: AdditionMonoidWithSub + One, V, W, X> IncDec
    for Vec<(T, U, V, W, X)>
{
    fn inc(mut self) -> Self {
        self.iter_mut().for_each(|(i, j, ..)| {
            *i += T::one();
            *j += U::one();
        });
        self
    }

    fn dec(mut self) -> Self {
        self.iter_mut().for_each(|(i, j, ..)| {
            *i -= T::one();
            *j -= U::one();
        });
        self
    }
}

impl<T: AdditionMonoidWithSub + One, U: AdditionMonoidWithSub + One> IncDec for (T, U) {
    fn inc(mut self) -> Self {
        self.0 += T::one();
        self.1 += U::one();
        self
    }

    fn dec(mut self) -> Self {
        self.0 -= T::one();
        self.1 -= U::one();
        self
    }
}
}
}
}
pub mod 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 flow_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;
use crate::algo_lib::graph::edges::flow_edge_trait::FlowEdgeTrait;
use crate::algo_lib::graph::graph::Graph;
use crate::algo_lib::numbers::num_traits::algebra::AdditionMonoidWithSub;

#[derive(Clone)]
pub struct FlowEdgeRaw<C: AdditionMonoidWithSub + PartialOrd + Copy, Id: EdgeId, P> {
    to: u32,
    capacity: C,
    reverse_id: u32,
    id: Id,
    payload: P,
}

impl<C: AdditionMonoidWithSub + PartialOrd + Copy, Id: EdgeId> FlowEdgeRaw<C, Id, ()> {
    pub fn new(from: usize, to: usize, c: C) -> (usize, Self) {
        (
            from,
            Self {
                to: to as u32,
                capacity: c,
                reverse_id: 0,
                id: Id::new(),
                payload: (),
            },
        )
    }
}

impl<C: AdditionMonoidWithSub + PartialOrd + Copy, Id: EdgeId, P> FlowEdgeRaw<C, Id, P> {
    pub fn with_payload(from: usize, to: usize, c: C, payload: P) -> (usize, Self) {
        (from, Self::with_payload_impl(to, c, payload))
    }

    fn with_payload_impl(to: usize, c: C, payload: P) -> Self {
        Self {
            to: to as u32,
            capacity: c,
            reverse_id: 0,
            id: Id::new(),
            payload,
        }
    }
}

impl<C: AdditionMonoidWithSub + PartialOrd + Copy, Id: EdgeId, P: Clone> EdgeTrait
    for FlowEdgeRaw<C, 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 {
        self.reverse_id as usize
    }

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

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

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

impl<C: AdditionMonoidWithSub + PartialOrd + Copy, Id: EdgeId, P: Clone> FlowEdgeTrait<C>
    for FlowEdgeRaw<C, Id, P>
{
    fn capacity(&self) -> C {
        self.capacity
    }

    fn capacity_mut(&mut self) -> &mut C {
        &mut self.capacity
    }

    fn flow(&self, graph: &Graph<Self>) -> C {
        graph[self.to as usize][self.reverse_id as usize].capacity
    }
}

pub type FlowEdge<C, P> = FlowEdgeRaw<C, NoId, P>;
pub type FlowEdgeWithId<C, P> = FlowEdgeRaw<C, WithId, P>;
}
pub mod flow_edge_trait {
use crate::algo_lib::graph::edges::edge_trait::EdgeTrait;
use crate::algo_lib::graph::graph::Graph;
use crate::algo_lib::numbers::num_traits::algebra::AdditionMonoidWithSub;

pub trait FlowEdgeTrait<C: AdditionMonoidWithSub + PartialOrd + Copy>: EdgeTrait {
    fn capacity(&self) -> C;
    fn capacity_mut(&mut self) -> &mut C;
    fn flow(&self, graph: &Graph<Self>) -> C;
    fn push_flow(&self, flow: C) -> (usize, usize, C) {
        (self.to(), self.reverse_id(), flow)
    }
}
}
}
pub mod flow_graph {
use crate::algo_lib::graph::edges::flow_edge_trait::FlowEdgeTrait;
use crate::algo_lib::graph::graph::Graph;
use crate::algo_lib::numbers::num_traits::algebra::AdditionMonoidWithSub;

pub trait FlowGraph<C: AdditionMonoidWithSub + PartialOrd + Copy, E: FlowEdgeTrait<C>> {
    fn push_flow(&mut self, push_data: (usize, usize, C));
}

impl<C: AdditionMonoidWithSub + PartialOrd + Copy, E: FlowEdgeTrait<C>> FlowGraph<C, E>
    for Graph<E>
{
    fn push_flow(&mut self, (to, reverse_id, flow): (usize, usize, C)) {
        *self.edges[to][reverse_id].capacity_mut() += flow;
        let from = self.edges[to][reverse_id].to();
        let direct_id = self.edges[to][reverse_id].reverse_id();
        let direct = &mut self.edges[from][direct_id];
        assert!(flow >= C::zero() && flow <= direct.capacity());
        *direct.capacity_mut() -= flow;
    }
}
}
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> {
    pub(super) 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.edges.len());
        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 max_flow {
use crate::algo_lib::collections::slice_ext::legacy_fill::LegacyFill;
use crate::algo_lib::graph::edges::flow_edge_trait::FlowEdgeTrait;
use crate::algo_lib::graph::flow_graph::FlowGraph;
use crate::algo_lib::graph::graph::Graph;
use crate::algo_lib::misc::recursive_function::Callable2;
use crate::algo_lib::misc::recursive_function::RecursiveFunction2;
use crate::algo_lib::numbers::num_traits::algebra::AdditionMonoidWithSub;
use crate::algo_lib::numbers::num_traits::ord::MinMax;
use crate::when;
use std::collections::VecDeque;

pub trait MaxFlow<C: AdditionMonoidWithSub + Ord + Copy + MinMax> {
    fn max_flow(&mut self, source: usize, destination: usize) -> C;
}

impl<C: AdditionMonoidWithSub + Ord + Copy + MinMax, E: FlowEdgeTrait<C>> MaxFlow<C> for Graph<E> {
    fn max_flow(&mut self, source: usize, destination: usize) -> C {
        let n = self.vertex_count();
        let mut dist = vec![0u32; n];
        let mut next_edge = vec![0u32; n];
        let inf = C::max_val();
        let mut total_flow = C::zero();
        let mut q = VecDeque::new();
        loop {
            // 1.43
            dist.legacy_fill(std::u32::MAX);
            dist[source] = 0;
            q.push_back(source);
            while !q.is_empty() {
                let cur = q.pop_front().unwrap();
                for e in self[cur].iter() {
                    if e.capacity() != C::zero() {
                        let next = e.to();
                        // 1.43
                        if dist[next] == std::u32::MAX {
                            dist[next] = dist[cur] + 1;
                            q.push_back(next);
                        }
                    }
                }
            }
            // 1.43
            if dist[destination] == std::u32::MAX {
                break;
            }
            next_edge.legacy_fill(0);
            let mut dinic_impl = RecursiveFunction2::new(|f, source, mut flow| -> C {
                when! {
                    source == destination => flow,
                    flow == C::zero() || dist[source] == dist[destination] => C::zero(),
                    else => {
                        let mut total_pushed = C::zero();
                        while (next_edge[source] as usize) < self[source].len() {
                            let edge = &self[source][next_edge[source] as usize];
                            if edge.capacity() != C::zero() && dist[edge.to()] == dist[source] + 1 {
                                let pushed = f.call(edge.to(), flow.min(edge.capacity()));
                                if pushed != C::zero() {
                                    let push_data = edge.push_flow(pushed);
                                    self.push_flow(push_data);
                                    flow -= pushed;
                                    total_pushed += pushed;
                                    if flow == C::zero() {
                                        return total_pushed;
                                    }
                                }
                            }
                            next_edge[source] += 1;
                        }
                        total_pushed
                    },
                }
            });
            total_flow += dinic_impl.call(source, inf);
        }
        total_flow
    }
}
}
}
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,
    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) -> 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, 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) -> char {
        self.skip_whitespace();
        self.get().unwrap().into()
    }

    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 char {
    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 u8 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,
}

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,
        }
    }

    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,
        }
    }

    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;
    }
}

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<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!(u8 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 when {
#[macro_export]
macro_rules! when {
    {$($cond: expr => $then: expr,)*} => {
        match () {
            $(_ if $cond => $then,)*
            _ => unreachable!(),
        }
    };
    {$($cond: expr => $then: expr,)* else $(=>)? $else: expr,} => {
        match () {
            $(_ if $cond => $then,)*
            _ => $else,
        }
    };
}
}
}
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::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)
    }

    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)
    }
}

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 ord {
pub trait MinMax: PartialOrd {
    fn min_val() -> Self;
    fn max_val() -> Self;
}

macro_rules! min_max_integer_impl {
    ($($t: ident)+) => {$(
        impl MinMax for $t {
            fn min_val() -> Self {
                // 1.43
                std::$t::MIN
            }

            fn max_val() -> Self {
                // 1.43
                std::$t::MAX
            }
        }
    )+};
}

min_max_integer_impl!(i128 i64 i32 i16 i8 isize u128 u64 u32 u16 u8 usize);
}
}
}
}
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);
}