// https://contest.ucup.ac/contest/1873/problem/9774
use crate::algo_lib::collections::segment_tree::{SegmentTree, SegmentTreeNode};
use crate::algo_lib::io::input::Input;
use crate::algo_lib::io::output::Output;
use crate::algo_lib::misc::random::random;
use crate::algo_lib::misc::test_type::TaskType;

use crate::algo_lib::misc::test_type::TestType;
use crate::algo_lib::misc::value_ref::ValueRef;
use crate::algo_lib::numbers::mod_int::ModInt;
use crate::algo_lib::numbers::num_traits::invertible::Invertible;
use crate::algo_lib::numbers::num_utils::Powers;
use crate::algo_lib::numbers::primes::prime::next_prime;
type PreCalc = ();
fn solve(input: &mut Input, out: &mut Output, _test_case: usize, _data: &mut PreCalc) {
    let n = input.read_size();
    let q = input.read_size();
    let a = input.read_size_vec(n);
    dynamic_value!(
        HM : i64 = next_prime(random().gen_range(10i64.pow(18)..= 2 * 10i64.pow(18)),)
    );
    type HashMod = ModInt<i64, HM>;
    let mult = HashMod::new(random().gen_range(4 * 10i64.pow(17)..=5 * 10i64.pow(17)));
    let rev = mult.inv().unwrap();
    value_ref!(Mult MULT : Powers < HashMod > = Powers::new(mult, 200_001));
    value_ref!(Rev REV : Powers < HashMod > = Powers::new(rev, 200_001));
    #[derive(Clone, Default)]
    struct Node {
        min: usize,
        max: usize,
        hash_up: HashMod,
        hash_down: HashMod,
        delta: usize,
    }
    impl Node {
        fn new(a: usize) -> Self {
            Self {
                min: a,
                max: a,
                hash_up: Mult::val().power(a),
                hash_down: Rev::val().power(a),
                delta: 0,
            }
        }
    }
    impl SegmentTreeNode for Node {
        fn new(_left: usize, _right: usize) -> Self {
            Self::default()
        }
        fn join(&mut self, left_val: &Self, right_val: &Self) {
            self.min = left_val.min.min(right_val.min);
            self.max = left_val.max.max(right_val.max);
            self.hash_up = left_val.hash_up + right_val.hash_up;
            self.hash_down = left_val.hash_down + right_val.hash_down;
        }
        fn accumulate(&mut self, value: &Self) {
            self.delta += value.delta;
            self.min += value.delta;
            self.max += value.delta;
            self.hash_up *= Mult::val().power(value.delta);
            self.hash_down *= Rev::val().power(value.delta);
        }
        fn reset_delta(&mut self) {
            self.delta = 0;
        }
    }
    let mut st = SegmentTree::gen(n, |i| Node::new(a[i]));
    for _ in 0..q {
        let command = input.read_int();
        match command {
            1 => {
                let l = input.read_size() - 1;
                let r = input.read_size();
                let v = input.read_size();
                st.update(
                    l..r,
                    &Node {
                        delta: v,
                        ..Default::default()
                    },
                );
            }
            2 => {
                let l = input.read_size() - 1;
                let r = input.read_size();
                let node = st.query(l..r);
                out.print_line(
                    node.hash_up * Rev::val().power(node.min)
                        == node.hash_down * Mult::val().power(node.max),
                );
            }
            _ => unreachable!(),
        }
    }
}
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,
    }
}


fn main() {
    let mut sin = std::io::stdin();
    let input = crate::algo_lib::io::input::Input::new(&mut sin);
    let mut stdout = std::io::stdout();
    let output = crate::algo_lib::io::output::Output::new(&mut stdout);
    run(input, output);
}
pub mod algo_lib {
pub mod collections {
pub mod bounds {
use std::ops::RangeBounds;
pub fn clamp(range: impl RangeBounds<usize>, n: usize) -> (usize, usize) {
    let start = match range.start_bound() {
        std::ops::Bound::Included(&x) => x,
        std::ops::Bound::Excluded(&x) => x + 1,
        std::ops::Bound::Unbounded => 0,
    };
    let end = match range.end_bound() {
        std::ops::Bound::Included(&x) => x + 1,
        std::ops::Bound::Excluded(&x) => x,
        std::ops::Bound::Unbounded => n,
    };
    (start, end.min(n))
}
}
pub mod fx_hash_map {
use std::cell::Cell;
use std::convert::TryInto;
use std::time::SystemTime;
use std::{
    collections::{HashMap, HashSet},
    hash::{BuildHasherDefault, Hasher},
    mem::size_of, 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 segment_tree {
use crate::algo_lib::collections::bounds::clamp;
use crate::algo_lib::misc::direction::Direction;
use crate::when;
use std::marker::PhantomData;
use std::ops::RangeBounds;
#[allow(unused_variables)]
pub trait SegmentTreeNode {
    fn new(left: usize, right: usize) -> Self;
    fn join(&mut self, left_val: &Self, right_val: &Self) {}
    fn accumulate(&mut self, value: &Self) {}
    fn reset_delta(&mut self) {}
}
pub trait Pushable<T>: SegmentTreeNode {
    fn push(&mut self, delta: T);
}
impl<T: SegmentTreeNode> Pushable<&T> for T {
    fn push(&mut self, delta: &T) {
        self.accumulate(delta);
    }
}
impl<T: SegmentTreeNode> Pushable<T> for T {
    fn push(&mut self, delta: T) {
        *self = delta;
    }
}
pub trait QueryResult<Result, Args>: SegmentTreeNode {
    fn empty_result(args: &Args) -> Result;
    fn result(&self, args: &Args) -> Result;
    fn join_results(
        left_res: Result,
        right_res: Result,
        args: &Args,
        left: usize,
        mid: usize,
        right: usize,
    ) -> Result;
}
impl<T: SegmentTreeNode + Clone> QueryResult<T, ()> for T {
    fn empty_result(_: &()) -> T {
        Self::new(0, 0)
    }
    fn result(&self, _: &()) -> T {
        self.clone()
    }
    fn join_results(
        left_res: T,
        right_res: T,
        _: &(),
        left: usize,
        mid: usize,
        right: usize,
    ) -> T {
        when! {
            left == mid => right_res, right == mid => left_res, else => { let mut res =
            Self::new(left, right); res.join(& left_res, & right_res); res },
        }
    }
}
#[derive(Clone)]
pub struct SegmentTree<Node> {
    n: usize,
    nodes: Vec<Node>,
}
impl<Node: SegmentTreeNode> SegmentTree<Node> {
    pub fn new(n: usize) -> Self {
        Self::gen(n, |left| Node::new(left, left + 1))
    }
    pub fn from_array(arr: Vec<Node>) -> Self {
        let n = arr.len();
        let mut iter = arr.into_iter();
        Self::gen(n, |_| iter.next().unwrap())
    }
    pub fn gen<F>(n: usize, gen: F) -> Self
    where
        F: FnMut(usize) -> Node,
    {
        if n == 0 {
            return Self {
                n,
                nodes: vec![Node::new(0, 0)],
            };
        }
        let mut res = Self {
            n,
            nodes: Vec::with_capacity(2 * n - 1),
        };
        res.init(gen);
        res
    }
    fn init<F>(&mut self, mut f: F)
    where
        F: FnMut(usize) -> Node,
    {
        self.init_impl(2 * self.n - 2, 0, self.n, &mut f);
    }
    fn init_impl<F>(&mut self, root: usize, left: usize, right: usize, f: &mut F)
    where
        F: FnMut(usize) -> Node,
    {
        if left + 1 == right {
            self.nodes.push(f(left));
        } else {
            let mid = left + ((right - left) >> 1);
            let left_child = root - 2 * (right - mid);
            let right_child = root - 1;
            self.init_impl(left_child, left, mid, f);
            self.init_impl(right_child, mid, right, f);
            let mut node = Node::new(left, right);
            node.join(&self.nodes[left_child], &self.nodes[right_child]);
            self.nodes.push(node);
        }
    }
    pub fn point_query(&mut self, at: usize) -> &Node {
        assert!(at < self.n);
        self.do_point_query(self.nodes.len() - 1, 0, self.n, at)
    }
    fn do_point_query(
        &mut self,
        root: usize,
        left: usize,
        right: usize,
        at: usize,
    ) -> &Node {
        if left + 1 == right {
            &self.nodes[root]
        } else {
            let mid = (left + right) >> 1;
            self.push_down(root, mid, right);
            let left_child = root - 2 * (right - mid);
            let right_child = root - 1;
            if at < mid {
                self.do_point_query(left_child, left, mid, at)
            } else {
                self.do_point_query(right_child, mid, right, at)
            }
        }
    }
    pub fn point_update<T>(&mut self, at: usize, val: T)
    where
        Node: Pushable<T>,
    {
        assert!(at < self.n);
        self.do_point_update(self.nodes.len() - 1, 0, self.n, at, val);
    }
    fn do_point_update<T>(
        &mut self,
        root: usize,
        left: usize,
        right: usize,
        at: usize,
        val: T,
    )
    where
        Node: Pushable<T>,
    {
        if left + 1 == right {
            self.nodes[root].push(val);
        } else {
            let mid = (left + right) >> 1;
            self.push_down(root, mid, right);
            let left_child = root - 2 * (right - mid);
            let right_child = root - 1;
            if at < mid {
                self.do_point_update(left_child, left, mid, at, val);
            } else {
                self.do_point_update(right_child, mid, right, at, val);
            }
            self.join(root, mid, right);
        }
    }
    pub fn point_through_update<'a, T>(&mut self, at: usize, val: &'a T)
    where
        Node: Pushable<&'a T>,
    {
        assert!(at < self.n);
        self.do_point_through_update(self.nodes.len() - 1, 0, self.n, at, val);
    }
    fn do_point_through_update<'a, T>(
        &mut self,
        root: usize,
        left: usize,
        right: usize,
        at: usize,
        val: &'a T,
    )
    where
        Node: Pushable<&'a T>,
    {
        self.nodes[root].push(val);
        if left + 1 != right {
            let mid = (left + right) >> 1;
            self.push_down(root, mid, right);
            let left_child = root - 2 * (right - mid);
            let right_child = root - 1;
            if at < mid {
                self.do_point_through_update(left_child, left, mid, at, val);
            } else {
                self.do_point_through_update(right_child, mid, right, at, val);
            }
        }
    }
    pub fn point_operation<Args, Res>(
        &mut self,
        op: &mut dyn PointOperation<Node, Args, Res>,
        args: Args,
    ) -> Res {
        assert_ne!(self.n, 0);
        self.do_point_operation(op, self.nodes.len() - 1, 0, self.n, args)
    }
    fn do_point_operation<Args, Res>(
        &mut self,
        op: &mut dyn PointOperation<Node, Args, Res>,
        root: usize,
        left: usize,
        right: usize,
        args: Args,
    ) -> Res {
        if left + 1 == right {
            op.adjust_leaf(&mut self.nodes[root], left, args)
        } else {
            let mid = (left + right) >> 1;
            self.push_down(root, mid, right);
            let left_child = root - 2 * (right - mid);
            let right_child = root - 1;
            let (l, r) = self.nodes.split_at_mut(root);
            let (l, m) = l.split_at_mut(right_child);
            let direction = op
                .select_branch(
                    &mut r[0],
                    &mut l[left_child],
                    &mut m[0],
                    &args,
                    left,
                    mid,
                    right,
                );
            let res = match direction {
                Direction::Left => {
                    self.do_point_operation(op, left_child, left, mid, args)
                }
                Direction::Right => {
                    self.do_point_operation(op, right_child, mid, right, args)
                }
            };
            self.join(root, mid, right);
            res
        }
    }
    pub fn update<'a, T>(&mut self, range: impl RangeBounds<usize>, val: &'a T)
    where
        Node: Pushable<&'a T>,
    {
        let (from, to) = clamp(range, self.n);
        self.do_update(self.nodes.len() - 1, 0, self.n, from, to, val)
    }
    pub fn do_update<'a, T>(
        &mut self,
        root: usize,
        left: usize,
        right: usize,
        from: usize,
        to: usize,
        val: &'a T,
    )
    where
        Node: Pushable<&'a T>,
    {
        when! {
            left >= to || right <= from => {}, left >= from && right <= to => self
            .nodes[root].push(val), else => { let mid = (left + right) >> 1; self
            .push_down(root, mid, right); let left_child = root - 2 * (right - mid); let
            right_child = root - 1; self.do_update(left_child, left, mid, from, to, val);
            self.do_update(right_child, mid, right, from, to, val); self.join(root, mid,
            right); },
        }
    }
    pub fn operation<Args, Res>(
        &mut self,
        range: impl RangeBounds<usize>,
        op: &mut dyn Operation<Node, Args, Res>,
        args: &Args,
    ) -> Res {
        let (from, to) = clamp(range, self.n);
        self.do_operation(self.nodes.len() - 1, 0, self.n, from, to, op, args)
    }
    pub fn do_operation<Args, Res>(
        &mut self,
        root: usize,
        left: usize,
        right: usize,
        from: usize,
        to: usize,
        op: &mut dyn Operation<Node, Args, Res>,
        args: &Args,
    ) -> Res {
        when! {
            left >= to || right <= from => op.empty_result(left, right, args), left >=
            from && right <= to => op.process_result(& mut self.nodes[root], args), else
            => { let mid = (left + right) >> 1; self.push_down(root, mid, right); let
            left_child = root - 2 * (right - mid); let right_child = root - 1; let
            left_result = self.do_operation(left_child, left, mid, from, to, op, args);
            let right_result = self.do_operation(right_child, mid, right, from, to, op,
            args); self.join(root, mid, right); op.join_results(left_result,
            right_result, args) },
        }
    }
    pub fn binary_search<Res>(
        &mut self,
        wh: impl FnMut(&Node, &Node) -> Direction,
        calc: impl FnMut(&Node, usize) -> Res,
    ) -> Res {
        self.do_binary_search(self.nodes.len() - 1, 0, self.n, wh, calc)
    }
    fn do_binary_search<Res>(
        &mut self,
        root: usize,
        left: usize,
        right: usize,
        mut wh: impl FnMut(&Node, &Node) -> Direction,
        mut calc: impl FnMut(&Node, usize) -> Res,
    ) -> Res {
        if left + 1 == right {
            calc(&self.nodes[root], left)
        } else {
            let mid = (left + right) >> 1;
            self.push_down(root, mid, right);
            let left_child = root - 2 * (right - mid);
            let right_child = root - 1;
            let direction = wh(&self.nodes[left_child], &self.nodes[right_child]);
            match direction {
                Direction::Left => self.do_binary_search(left_child, left, mid, wh, calc),
                Direction::Right => {
                    self.do_binary_search(right_child, mid, right, wh, calc)
                }
            }
        }
    }
    fn join(&mut self, root: usize, mid: usize, right: usize) {
        let left_child = root - 2 * (right - mid);
        let right_child = root - 1;
        let (left_node, right_node) = self.nodes.split_at_mut(root);
        right_node[0].join(&left_node[left_child], &left_node[right_child]);
    }
    fn do_push_down(&mut self, parent: usize, to: usize) {
        let (left_nodes, right_nodes) = self.nodes.split_at_mut(parent);
        left_nodes[to].accumulate(&right_nodes[0]);
    }
    fn push_down(&mut self, root: usize, mid: usize, right: usize) {
        self.do_push_down(root, root - 2 * (right - mid));
        self.do_push_down(root, root - 1);
        self.nodes[root].reset_delta();
    }
    pub fn query<T>(&mut self, range: impl RangeBounds<usize>) -> T
    where
        Node: QueryResult<T, ()>,
    {
        let (from, to) = clamp(range, self.n);
        if from >= to {
            Node::empty_result(&())
        } else {
            self.do_query(self.nodes.len() - 1, 0, self.n, from, to, &())
        }
    }
    pub fn query_with_args<T, Args>(
        &mut self,
        range: impl RangeBounds<usize>,
        args: &Args,
    ) -> T
    where
        Node: QueryResult<T, Args>,
    {
        let (from, to) = clamp(range, self.n);
        if from >= to {
            Node::empty_result(args)
        } else {
            self.do_query(self.nodes.len() - 1, 0, self.n, from, to, args)
        }
    }
    fn do_query<T, Args>(
        &mut self,
        root: usize,
        left: usize,
        right: usize,
        from: usize,
        to: usize,
        args: &Args,
    ) -> T
    where
        Node: QueryResult<T, Args>,
    {
        if left >= from && right <= to {
            self.nodes[root].result(args)
        } else {
            let mid = (left + right) >> 1;
            self.push_down(root, mid, right);
            let left_child = root - 2 * (right - mid);
            let right_child = root - 1;
            when! {
                to <= mid => self.do_query(left_child, left, mid, from, to, args), from
                >= mid => self.do_query(right_child, mid, right, from, to, args), else =>
                { let left_result = self.do_query(left_child, left, mid, from, to, args);
                let right_result = self.do_query(right_child, mid, right, from, to,
                args); Node::join_results(left_result, right_result, args, left, mid,
                right) },
            }
        }
    }
}
pub trait PointOperation<Node: SegmentTreeNode, Args, Res = Node> {
    fn adjust_leaf(&mut self, leaf: &mut Node, at: usize, args: Args) -> Res;
    fn select_branch(
        &mut self,
        root: &mut Node,
        left_child: &mut Node,
        right_child: &mut Node,
        args: &Args,
        left: usize,
        mid: usize,
        right: usize,
    ) -> Direction;
}
pub struct PointOperationClosure<'s, Node: SegmentTreeNode, Args, Res = Node> {
    adjust_leaf: Box<dyn FnMut(&mut Node, usize, Args) -> Res + 's>,
    select_branch: Box<
        dyn FnMut(
            &mut Node,
            &mut Node,
            &mut Node,
            &Args,
            usize,
            usize,
            usize,
        ) -> Direction + 's,
    >,
    phantom_node: PhantomData<Node>,
    phantom_args: PhantomData<Args>,
    phantom_res: PhantomData<Res>,
}
impl<'s, Node: SegmentTreeNode, Args, Res> PointOperationClosure<'s, Node, Args, Res> {
    pub fn new<F1, F2>(adjust_leaf: F1, select_branch: F2) -> Self
    where
        F1: FnMut(&mut Node, usize, Args) -> Res + 's,
        F2: FnMut(
                &mut Node,
                &mut Node,
                &mut Node,
                &Args,
                usize,
                usize,
                usize,
            ) -> Direction + 's,
    {
        Self {
            adjust_leaf: Box::new(adjust_leaf),
            select_branch: Box::new(select_branch),
            phantom_node: Default::default(),
            phantom_args: Default::default(),
            phantom_res: Default::default(),
        }
    }
}
impl<'s, Node: SegmentTreeNode, Args, Res> PointOperation<Node, Args, Res>
for PointOperationClosure<'s, Node, Args, Res> {
    fn adjust_leaf(&mut self, leaf: &mut Node, at: usize, args: Args) -> Res {
        (self.adjust_leaf)(leaf, at, args)
    }
    fn select_branch(
        &mut self,
        root: &mut Node,
        left_child: &mut Node,
        right_child: &mut Node,
        args: &Args,
        left: usize,
        mid: usize,
        right: usize,
    ) -> Direction {
        (self.select_branch)(root, left_child, right_child, args, left, mid, right)
    }
}
pub trait Operation<Node: SegmentTreeNode, Args, Res = Node> {
    fn process_result(&mut self, node: &mut Node, args: &Args) -> Res;
    fn join_results(&mut self, left_res: Res, right_res: Res, args: &Args) -> Res;
    fn empty_result(&mut self, left: usize, right: usize, args: &Args) -> Res;
}
pub struct OperationClosure<'s, Node: SegmentTreeNode, Args, Res = Node> {
    process_result: Box<dyn FnMut(&mut Node, &Args) -> Res + 's>,
    join_results: Box<dyn FnMut(Res, Res, &Args) -> Res + 's>,
    empty_result: Box<dyn FnMut(usize, usize, &Args) -> Res + 's>,
    phantom_node: PhantomData<Node>,
    phantom_args: PhantomData<Args>,
    phantom_res: PhantomData<Res>,
}
impl<'s, Node: SegmentTreeNode, Args, Res> OperationClosure<'s, Node, Args, Res> {
    pub fn new<F1, F2, F3>(
        process_result: F1,
        join_results: F2,
        empty_result: F3,
    ) -> Self
    where
        F1: FnMut(&mut Node, &Args) -> Res + 's,
        F2: FnMut(Res, Res, &Args) -> Res + 's,
        F3: FnMut(usize, usize, &Args) -> Res + 's,
    {
        Self {
            process_result: Box::new(process_result),
            join_results: Box::new(join_results),
            empty_result: Box::new(empty_result),
            phantom_node: Default::default(),
            phantom_args: Default::default(),
            phantom_res: Default::default(),
        }
    }
}
impl<'s, Node: SegmentTreeNode, Args, Res> Operation<Node, Args, Res>
for OperationClosure<'s, Node, Args, Res> {
    fn process_result(&mut self, node: &mut Node, args: &Args) -> Res {
        (self.process_result)(node, args)
    }
    fn join_results(&mut self, left_res: Res, right_res: Res, args: &Args) -> Res {
        (self.join_results)(left_res, right_res, args)
    }
    fn empty_result(&mut self, left: usize, right: usize, args: &Args) -> Res {
        (self.empty_result)(left, right, args)
    }
}
}
pub mod slice_ext {
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 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 io {
pub mod input {
use crate::algo_lib::collections::vec_ext::default::default_vec;
use std::io::Read;
use std::mem::MaybeUninit;
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) }
    }
    pub fn is_exhausted(&mut self) -> bool {
        self.peek().is_none()
    }
    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)
    }
}
impl<T: Readable, const SIZE: usize> Readable for [T; SIZE] {
    fn read(input: &mut Input) -> Self {
        unsafe {
            let mut res = MaybeUninit::<[T; SIZE]>::uninit();
            for i in 0..SIZE {
                let ptr: *mut T = (*res.as_mut_ptr()).as_mut_ptr();
                ptr.add(i).write(input.read::<T>());
            }
            res.assume_init()
        }
    }
}
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, Stderr, 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>,
    separator: u8,
}
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,
            separator: b' ',
        }
    }
    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,
            separator: b' ',
        }
    }
    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(self.separator);
            }
            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: usize) {
        self.precision = Some(precision);
    }
    pub fn reset_precision(&mut self) {
        self.precision = None;
    }
    pub fn get_precision(&self) -> Option<usize> {
        self.precision
    }
    pub fn separator(&self) -> u8 {
        self.separator
    }
    pub fn set_separator(&mut self, separator: u8) {
        self.separator = separator;
    }
}
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); $(out.put(out
        .separator); self.$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 direction {
#[derive(Copy, Clone)]
pub enum Direction {
    Left,
    Right,
}
}
pub mod random {
use crate::algo_lib::collections::slice_ext::indices::Indices;
use crate::algo_lib::numbers::num_traits::algebra::IntegerSemiRingWithSub;
use crate::algo_lib::numbers::num_traits::ord::MinMax;
use crate::algo_lib::numbers::num_traits::primitive::Primitive;
use std::ops::{RangeBounds, Rem};
use std::time::SystemTime;
const NN: usize = 312;
const MM: usize = 156;
const MATRIX_A: u64 = 0xB5026F5AA96619E9;
const UM: u64 = 0xFFFFFFFF80000000;
const LM: u64 = 0x7FFFFFFF;
const F: u64 = 6364136223846793005;
const MAG01: [u64; 2] = [0, MATRIX_A];
pub struct Random {
    mt: [u64; NN],
    index: usize,
}
impl Random {
    pub fn new(seed: u64) -> Self {
        let mut res = Self { mt: [0u64; NN], index: NN };
        res.mt[0] = seed;
        for i in 1..NN {
            res.mt[i] = F
                .wrapping_mul(res.mt[i - 1] ^ (res.mt[i - 1] >> 62))
                .wrapping_add(i as u64);
        }
        res
    }
    fn gen_impl(&mut self) -> u64 {
        if self.index == NN {
            for i in 0..(NN - MM) {
                let x = (self.mt[i] & UM) | (self.mt[i + 1] & LM);
                self.mt[i] = self.mt[i + MM] ^ (x >> 1) ^ MAG01[(x & 1) as usize];
            }
            for i in (NN - MM)..(NN - 1) {
                let x = (self.mt[i] & UM) | (self.mt[i + 1] & LM);
                self.mt[i] = self.mt[i + MM - NN] ^ (x >> 1) ^ MAG01[(x & 1) as usize];
            }
            let x = (self.mt[NN - 1] & UM) | (self.mt[0] & LM);
            self.mt[NN - 1] = self.mt[MM - 1] ^ (x >> 1) ^ MAG01[(x & 1) as usize];
            self.index = 0;
        }
        let mut x = self.mt[self.index];
        self.index += 1;
        x ^= (x >> 29) & 0x5555555555555555;
        x ^= (x << 17) & 0x71D67FFFEDA60000;
        x ^= (x << 37) & 0xFFF7EEE000000000;
        x ^= x >> 43;
        x
    }
    pub fn gen<T>(&mut self) -> T
    where
        u64: Primitive<T>,
    {
        self.gen_impl().to()
    }
    pub fn gen_u128(&mut self) -> u128 {
        (self.gen_impl() as u128) << 64 | self.gen_impl() as u128
    }
    pub fn gen_i128(&mut self) -> i128 {
        self.gen_u128() as i128
    }
    pub fn gen_bool(&mut self) -> bool {
        (self.gen_impl() & 1) == 1
    }
    pub fn gen_bound<T: Rem<Output = T> + Primitive<u64>>(&mut self, n: T) -> T
    where
        u64: Primitive<T>,
    {
        (self.gen_impl() % n.to()).to()
    }
    pub fn gen_range<T: IntegerSemiRingWithSub + Primitive<u64> + MinMax>(
        &mut self,
        range: impl RangeBounds<T>,
    ) -> T
    where
        u64: Primitive<T>,
    {
        let f = match range.start_bound() {
            std::ops::Bound::Included(&s) => s,
            std::ops::Bound::Excluded(&s) => s + T::one(),
            std::ops::Bound::Unbounded => T::min_val(),
        };
        let t = match range.end_bound() {
            std::ops::Bound::Included(&e) => e,
            std::ops::Bound::Excluded(&e) => e - T::one(),
            std::ops::Bound::Unbounded => T::max_val(),
        };
        if f == T::min_val() && t == T::max_val() {
            self.gen()
        } else {
            f + self.gen_bound(t - f + T::one())
        }
    }
}
static mut RAND: Option<Random> = None;
pub fn random() -> &'static mut Random {
    unsafe {
        if RAND.is_none() {
            RAND = Some(
                Random::new(
                    (SystemTime::UNIX_EPOCH.elapsed().unwrap().as_nanos()
                        & 0xFFFFFFFFFFFFFFFF) as u64,
                ),
            );
        }
        RAND.as_mut().unwrap()
    }
}
pub trait Shuffle {
    fn shuffle(&mut self);
}
impl<T> Shuffle for [T] {
    fn shuffle(&mut self) {
        for i in self.indices() {
            let at = random().gen_bound(i + 1);
            self.swap(i, at);
        }
    }
}
}
pub mod test_type {
pub enum TestType {
    Single,
    MultiNumber,
    MultiEof,
}
pub enum TaskType {
    Classic,
    Interactive,
}
}
pub mod value {
use std::hash::Hash;
pub trait Value<T>: Copy + Eq + Hash {
    fn val() -> T;
}
pub trait ConstValue<T>: Value<T> {
    const VAL: T;
}
impl<T, V: ConstValue<T>> Value<T> for V {
    fn val() -> T {
        Self::VAL
    }
}
#[macro_export]
macro_rules! value {
    ($name:ident : $t:ty = $val:expr) => {
        #[derive(Copy, Clone, Eq, PartialEq, Hash, Ord, PartialOrd, Default)] pub struct
        $name {} impl $crate ::algo_lib::misc::value::ConstValue <$t > for $name { const
        VAL : $t = $val; }
    };
}
pub trait DynamicValue<T>: Value<T> {
    fn set_val(t: T);
}
#[macro_export]
macro_rules! dynamic_value {
    ($name:ident : $t:ty, $val:ident) => {
        static mut $val : Option <$t > = None; #[derive(Copy, Clone, Eq, PartialEq, Hash,
        Default)] struct $name {} impl $crate ::algo_lib::misc::value::DynamicValue <$t >
        for $name { fn set_val(t : $t) { unsafe { $val = Some(t); } } } impl $crate
        ::algo_lib::misc::value::Value <$t > for $name { fn val() -> $t { unsafe { $val
        .unwrap() } } }
    };
    ($name:ident : $t:ty) => {
        dynamic_value!($name : $t, VAL);
    };
    ($name:ident : $t:ty = $val:expr) => {
        dynamic_value!($name : $t); <$name as $crate
        ::algo_lib::misc::value::DynamicValue <$t >>::set_val($val);
    };
    ($name:ident : $t:ty = $val:expr, $val_static:ident) => {
        dynamic_value!($name : $t, $val_static); <$name as $crate
        ::algo_lib::misc::value::DynamicValue <$t >>::set_val($val);
    };
}
}
pub mod value_ref {
pub trait ConstValueRef<T: ?Sized + 'static> {
    fn val() -> &'static T;
}
pub trait ValueRef<T: 'static> {
    fn val() -> &'static T;
    fn set_val(t: T);
    fn val_mut() -> &'static mut T;
}
#[macro_export]
macro_rules! const_value_ref {
    ($name:ident $val_name:ident : $int_t:ty as $ext_t:ty = $base:expr) => {
        const $val_name : $int_t = $base; #[derive(Copy, Clone, Eq, PartialEq, Hash)] pub
        struct $name {} impl $crate ::algo_lib::misc::value_ref::ConstValueRef <$ext_t >
        for $name { fn val() -> &'static $ext_t { &$val_name } }
    };
    ($name:ident $val_name:ident : $int_t:ty = $base:expr) => {
        const_value_ref!($name $val_name : $int_t as $int_t = $base);
    };
}
#[macro_export]
macro_rules! value_ref {
    ($name:ident $val_name:ident : $t:ty) => {
        static mut $val_name : Option <$t > = None; #[derive(Copy, Clone, Eq, PartialEq,
        Hash)] struct $name {} impl $crate ::algo_lib::misc::value_ref::ValueRef <$t >
        for $name { fn val() -> &'static $t { unsafe { $val_name .as_ref().unwrap() } }
        fn set_val(t : $t) { unsafe { $val_name = Some(t); } } fn val_mut() -> &'static
        mut $t { unsafe { $val_name .as_mut().unwrap() } } }
    };
    ($name:ident $val_name:ident : $t:ty = $init_val:expr) => {
        value_ref!($name $val_name : $t); <$name as $crate
        ::algo_lib::misc::value_ref::ValueRef <$t >>::set_val($init_val);
    };
}
}
pub mod when {
#[macro_export]
macro_rules! when {
    {$($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 gcd {
use crate::algo_lib::numbers::num_traits::algebra::{
    IntegerMultiplicationMonoid, IntegerSemiRingWithSub, Zero,
};
use std::mem::swap;
pub fn extended_gcd<T: IntegerSemiRingWithSub + Copy>(a: T, b: T) -> (T, T, T) {
    if a == T::zero() {
        (b, T::zero(), T::one())
    } else {
        let (d, y, mut x) = extended_gcd(b % a, a);
        x -= b / a * y;
        (d, x, y)
    }
}
pub fn gcd<T: Copy + Zero + IntegerMultiplicationMonoid>(mut a: T, mut b: T) -> T {
    while b != T::zero() {
        a %= b;
        swap(&mut a, &mut b);
    }
    a
}
pub fn lcm<T: Copy + Zero + IntegerMultiplicationMonoid>(a: T, b: T) -> T {
    (a / gcd(a, b)) * b
}
pub fn remainder<T: IntegerSemiRingWithSub + Copy>(
    a1: T,
    n1: T,
    a2: T,
    n2: T,
) -> Option<T> {
    let (d, m1, m2) = extended_gcd(n1, n2);
    if (a2 - a1) % d != T::zero() {
        return None;
    }
    let m = lcm(n1, n2);
    Some(((a1 * m2) % n1 * n2 + (a2 * m1) % n2 * n1) % m)
}
}
pub mod mod_int {
use crate::algo_lib::collections::fx_hash_map::FxHashMap;
use crate::algo_lib::io::input::{Input, Readable};
use crate::algo_lib::io::output::{Output, Writable};
use crate::algo_lib::misc::value::Value;
use crate::algo_lib::numbers::gcd::extended_gcd;
use crate::algo_lib::numbers::num_traits::algebra::{Field, IntegerRing, One, Ring, Zero};
use crate::algo_lib::numbers::num_traits::as_index::AsIndex;
use crate::algo_lib::numbers::num_traits::invertible::Invertible;
use crate::algo_lib::numbers::num_traits::wideable::Wideable;
use crate::{value, when};
use std::fmt::{Display, Formatter};
use std::hash::Hash;
use std::marker::PhantomData;
use std::ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Sub, SubAssign};
pub trait BaseModInt: Field + Copy {
    type W: IntegerRing + Copy + From<Self::T>;
    type T: IntegerRing + Ord + Copy + Wideable<W = Self::W>;
    fn from(v: Self::T) -> Self;
    fn module() -> Self::T;
    fn value(&self) -> Self::T;
}
#[derive(Copy, Clone, Eq, PartialEq, Hash, Default)]
pub struct ModInt<T, V: Value<T>> {
    n: T,
    phantom: PhantomData<V>,
}
impl<T: Ring + Ord + Copy, V: Value<T>> ModInt<T, V> {
    unsafe fn unchecked_new(n: T) -> Self {
        debug_assert!(n >= T::zero() && n < V::val());
        Self {
            n,
            phantom: Default::default(),
        }
    }
    unsafe fn maybe_subtract_mod(mut n: T) -> T {
        debug_assert!(n < V::val() + V::val() && n >= T::zero());
        if n >= V::val() {
            n -= V::val();
        }
        n
    }
}
impl<T: IntegerRing + Ord + Copy, V: Value<T>> ModInt<T, V> {
    pub fn new(n: T) -> Self {
        unsafe {
            Self::unchecked_new(Self::maybe_subtract_mod(n % (V::val()) + V::val()))
        }
    }
    pub fn val(&self) -> T {
        self.n
    }
}
impl<T: Copy + IntegerRing + Ord + Wideable + Hash, V: Value<T>> ModInt<T, V>
where
    T::W: Copy + IntegerRing,
{
    pub fn log(&self, alpha: Self) -> T {
        let mut base = FxHashMap::default();
        let mut exp = T::zero();
        let mut pow = Self::one();
        let mut inv = *self;
        let alpha_inv = alpha.inv().unwrap();
        while exp * exp < Self::module() {
            if inv == Self::one() {
                return exp;
            }
            base.insert(inv, exp);
            exp += T::one();
            pow *= alpha;
            inv *= alpha_inv;
        }
        let step = pow;
        let mut i = T::one();
        loop {
            if let Some(b) = base.get(&pow) {
                break exp * i + *b;
            }
            pow *= step;
            i += T::one();
        }
    }
}
impl<T: Wideable + Ring + Ord + Copy, V: Value<T>> ModInt<T, V>
where
    T::W: IntegerRing,
{
    pub fn new_from_wide(n: T::W) -> Self {
        unsafe {
            Self::unchecked_new(
                Self::maybe_subtract_mod(T::downcast(n % V::val().into()) + V::val()),
            )
        }
    }
}
impl<T: Copy + IntegerRing + Ord + Wideable, V: Value<T>> Invertible for ModInt<T, V>
where
    T::W: Copy + IntegerRing,
{
    type Output = Self;
    fn inv(&self) -> Option<Self> {
        let (g, x, _) = extended_gcd(T::W::from(self.n), T::W::from(V::val()));
        if g != T::W::one() { None } else { Some(Self::new_from_wide(x)) }
    }
}
impl<T: IntegerRing + Ord + Copy + Wideable, V: Value<T>> BaseModInt for ModInt<T, V>
where
    T::W: IntegerRing + Copy,
{
    type W = T::W;
    type T = T;
    fn from(v: Self::T) -> Self {
        Self::new(v)
    }
    fn module() -> T {
        V::val()
    }
    fn value(&self) -> T {
        self.n
    }
}
impl<T: IntegerRing + Ord + Copy, V: Value<T>> From<T> for ModInt<T, V> {
    fn from(n: T) -> Self {
        Self::new(n)
    }
}
impl<T: Ring + Ord + Copy, V: Value<T>> AddAssign for ModInt<T, V> {
    fn add_assign(&mut self, rhs: Self) {
        self.n = unsafe { Self::maybe_subtract_mod(self.n + rhs.n) };
    }
}
impl<T: Ring + Ord + Copy, V: Value<T>> Add for ModInt<T, V> {
    type Output = Self;
    fn add(mut self, rhs: Self) -> Self::Output {
        self += rhs;
        self
    }
}
impl<T: Ring + Ord + Copy, V: Value<T>> SubAssign for ModInt<T, V> {
    fn sub_assign(&mut self, rhs: Self) {
        self.n = unsafe { Self::maybe_subtract_mod(self.n + V::val() - rhs.n) };
    }
}
impl<T: Ring + Ord + Copy, V: Value<T>> Sub for ModInt<T, V> {
    type Output = Self;
    fn sub(mut self, rhs: Self) -> Self::Output {
        self -= rhs;
        self
    }
}
impl<T: IntegerRing + Ord + Copy + Wideable, V: Value<T>> MulAssign for ModInt<T, V>
where
    T::W: IntegerRing + Copy,
{
    fn mul_assign(&mut self, rhs: Self) {
        self.n = T::downcast(
            T::W::from(self.n) * T::W::from(rhs.n) % T::W::from(V::val()),
        );
    }
}
impl<T: IntegerRing + Ord + Copy + Wideable, V: Value<T>> Mul for ModInt<T, V>
where
    T::W: IntegerRing + Copy,
{
    type Output = Self;
    fn mul(mut self, rhs: Self) -> Self::Output {
        self *= rhs;
        self
    }
}
impl<T: IntegerRing + Ord + Copy + Wideable, V: Value<T>> DivAssign for ModInt<T, V>
where
    T::W: IntegerRing + Copy,
{
    #[allow(clippy::suspicious_op_assign_impl)]
    fn div_assign(&mut self, rhs: Self) {
        *self *= rhs.inv().unwrap();
    }
}
impl<T: IntegerRing + Ord + Copy + Wideable, V: Value<T>> Div for ModInt<T, V>
where
    T::W: IntegerRing + Copy,
{
    type Output = Self;
    fn div(mut self, rhs: Self) -> Self::Output {
        self /= rhs;
        self
    }
}
impl<T: Ring + Ord + Copy, V: Value<T>> Neg for ModInt<T, V> {
    type Output = Self;
    fn neg(mut self) -> Self::Output {
        self.n = unsafe { Self::maybe_subtract_mod(V::val() - self.n) };
        self
    }
}
impl<T: Display, V: Value<T>> Display for ModInt<T, V> {
    fn fmt(&self, f: &mut Formatter<'_>) -> std::fmt::Result {
        <T as Display>::fmt(&self.n, f)
    }
}
impl<T: IntegerRing + Ord + Copy + Readable, V: Value<T>> Readable for ModInt<T, V> {
    fn read(input: &mut Input) -> Self {
        Self::new(T::read(input))
    }
}
impl<T: Writable, V: Value<T>> Writable for ModInt<T, V> {
    fn write(&self, output: &mut Output) {
        self.n.write(output);
    }
}
impl<T: Ring + Ord + Copy, V: Value<T>> Zero for ModInt<T, V> {
    fn zero() -> Self {
        unsafe { Self::unchecked_new(T::zero()) }
    }
}
impl<T: IntegerRing + Ord + Copy, V: Value<T>> One for ModInt<T, V> {
    fn one() -> Self {
        Self::new(T::one())
    }
}
impl<T, V: Value<T>> Wideable for ModInt<T, V> {
    type W = Self;
    fn downcast(w: Self::W) -> Self {
        w
    }
}
impl<
    T: IntegerRing + Ord + Copy + Wideable + Display + AsIndex,
    V: Value<T>,
> std::fmt::Debug for ModInt<T, V>
where
    T::W: IntegerRing + Copy,
{
    fn fmt(&self, f: &mut Formatter) -> std::fmt::Result {
        let max = T::from_index(100);
        when! {
            self.n <= max => write!(f, "{}", self.n), self.n >= V::val() - max =>
            write!(f, "{}", self.n - V::val()), else => { let mut denominator = T::one();
            while denominator < max { let mut num = T::one(); while num < max { if
            Self::new(num) / Self::new(denominator) == * self { return write!(f, "{}/{}",
            num, denominator); } if - Self::new(num) / Self::new(denominator) == * self {
            return write!(f, "-{}/{}", num, denominator); } num += T::one(); }
            denominator += T::one(); } write!(f, "(?? {} ??)", self.n) },
        }
    }
}
impl<T: IntegerRing + Ord + Copy + AsIndex + Wideable, V: Value<T>> AsIndex
for ModInt<T, V>
where
    T::W: AsIndex + IntegerRing + Ord,
{
    fn from_index(idx: usize) -> Self {
        let t = T::W::from_index(idx);
        if t >= T::W::from(V::val()) {
            Self::new_from_wide(t)
        } else {
            unsafe { Self::unchecked_new(T::downcast(t)) }
        }
    }
    fn to_index(self) -> usize {
        self.n.to_index()
    }
}
value!(Val7 : i32 = 1_000_000_007);
pub type ModInt7 = ModInt<i32, Val7>;
value!(Val9 : i32 = 1_000_000_009);
pub type ModInt9 = ModInt<i32, Val9>;
value!(ValF : i32 = 998_244_353);
pub type ModIntF = ModInt<i32, ValF>;
}
pub mod num_traits {
pub mod algebra {
use crate::algo_lib::numbers::num_traits::invertible::Invertible;
use std::ops::{
    Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Rem, RemAssign, Sub, 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 $t { fn one() -> Self
        { 1 } })+
    };
}
zero_one_integer_impl!(i128 i64 i32 i16 i8 isize u128 u64 u32 u16 u8 usize);
}
pub mod as_index {
pub trait AsIndex {
    fn from_index(idx: usize) -> Self;
    fn to_index(self) -> usize;
}
macro_rules! from_index_impl {
    ($($t:ident)+) => {
        $(impl AsIndex for $t { fn from_index(idx : usize) -> Self { idx as $t } fn
        to_index(self) -> usize { self as usize } })+
    };
}
from_index_impl!(i128 i64 i32 i16 i8 isize u128 u64 u32 u16 u8 usize);
}
pub mod invertible {
pub trait Invertible {
    type Output;
    fn inv(&self) -> Option<Self::Output>;
}
}
pub mod ord {
pub trait MinMax: PartialOrd {
    fn min_val() -> Self;
    fn max_val() -> Self;
}
macro_rules! min_max_integer_impl {
    ($($t:ident)+) => {
        $(impl MinMax for $t { fn min_val() -> Self { std::$t ::MIN } fn max_val() ->
        Self { std::$t ::MAX } })+
    };
}
min_max_integer_impl!(i128 i64 i32 i16 i8 isize u128 u64 u32 u16 u8 usize);
}
pub mod primitive {
pub trait Primitive<T>: Copy {
    fn to(self) -> T;
}
macro_rules! primitive_one {
    ($t:ident, $($u:ident)+) => {
        $(impl Primitive <$u > for $t { fn to(self) -> $u { self as $u } })+
    };
}
macro_rules! primitive {
    ($($t:ident)+) => {
        $(primitive_one!($t, u8 u16 u32 u64 u128 usize i8 i16 i32 i64 i128 isize);)+
    };
}
primitive!(u8 u16 u32 u64 u128 usize i8 i16 i32 i64 i128 isize);
}
pub mod wideable {
use std::convert::From;
pub trait Wideable: Sized {
    type W: From<Self>;
    fn downcast(w: Self::W) -> Self;
}
macro_rules! wideable_impl {
    ($($t:ident $w:ident),+) => {
        $(impl Wideable for $t { type W = $w; fn downcast(w : Self::W) -> Self { w as $t
        } })+
    };
}
wideable_impl!(i64 i128, i32 i64, i16 i32, i8 i16, u64 u128, u32 u64, u16 u32, u8 u16);
}
}
pub mod num_utils {
use crate::algo_lib::numbers::num_traits::algebra::{
    AdditionMonoid, IntegerRing, MultiplicationMonoid,
};
use crate::algo_lib::numbers::num_traits::as_index::AsIndex;
pub fn factorials<T: MultiplicationMonoid + Copy + AsIndex>(len: usize) -> Vec<T> {
    let mut res = Vec::new();
    if len > 0 {
        res.push(T::one());
    }
    while res.len() < len {
        res.push((*res.last().unwrap()) * T::from_index(res.len()));
    }
    res
}
pub fn powers<T: MultiplicationMonoid + Copy>(base: T, len: usize) -> Vec<T> {
    let mut res = Vec::new();
    if len > 0 {
        res.push(T::one());
    }
    while res.len() < len {
        res.push((*res.last().unwrap()) * base);
    }
    res
}
pub struct Powers<T: MultiplicationMonoid + Copy> {
    small: Vec<T>,
    big: Vec<T>,
}
impl<T: MultiplicationMonoid + Copy> Powers<T> {
    pub fn new(base: T, len: usize) -> Self {
        let small = powers(base, len);
        let big = powers(small[len - 1] * base, len);
        Self { small, big }
    }
    pub fn power(&self, exp: usize) -> T {
        debug_assert!(exp < self.small.len() * self.small.len());
        self.big[exp / self.small.len()] * self.small[exp % self.small.len()]
    }
}
pub fn factorial<T: MultiplicationMonoid + AsIndex>(n: usize) -> T {
    let mut res = T::one();
    for i in 1..=n {
        res *= T::from_index(i);
    }
    res
}
pub trait PartialSums<T> {
    fn partial_sums(&self) -> Vec<T>;
}
impl<T: AdditionMonoid + Copy> PartialSums<T> for [T] {
    fn partial_sums(&self) -> Vec<T> {
        let mut res = Vec::with_capacity(self.len() + 1);
        res.push(T::zero());
        for i in self.iter() {
            res.push(*res.last().unwrap() + *i);
        }
        res
    }
}
pub trait UpperDiv {
    fn upper_div(self, other: Self) -> Self;
}
impl<T: IntegerRing + Copy> UpperDiv for T {
    fn upper_div(self, other: Self) -> Self {
        (self + other - Self::one()) / other
    }
}
}
pub mod number_ext {
use crate::algo_lib::numbers::num_traits::algebra::{
    IntegerSemiRing, MultiplicationMonoid,
};
use crate::algo_lib::numbers::num_traits::as_index::AsIndex;
use std::ops::Mul;
pub trait Power {
    #[must_use]
    fn power<T: IntegerSemiRing + Copy>(&self, exp: T) -> Self;
}
impl<S: MultiplicationMonoid + Copy> Power for S {
    fn power<T: IntegerSemiRing + Copy>(&self, exp: T) -> Self {
        if exp == T::zero() {
            S::one()
        } else {
            let mut res = self.power(exp / (T::one() + T::one()));
            res *= res;
            if exp % (T::one() + T::one()) == T::one() {
                res *= *self;
            }
            res
        }
    }
}
pub trait Digits<S> {
    fn num_digs(&self) -> usize;
    fn sum_digs(&self) -> Self;
    fn digits(&self) -> impl Iterator<Item = S>;
}
impl<S: IntegerSemiRing + AsIndex + Copy> Digits<S> for S {
    fn num_digs(&self) -> usize {
        let mut copy = *self;
        let ten = S::from_index(10);
        let mut res = 0;
        while copy != S::zero() {
            copy /= ten;
            res += 1;
        }
        res
    }
    fn sum_digs(&self) -> S {
        let mut copy = *self;
        let ten = S::from_index(10);
        let mut res = S::zero();
        while copy != S::zero() {
            res += copy % ten;
            copy /= ten;
        }
        res
    }
    fn digits(&self) -> impl Iterator<Item = Self> {
        let mut copy = *self;
        let ten = S::from_index(10);
        std::iter::from_fn(move || {
            if copy == S::zero() {
                None
            } else {
                let res = copy % ten;
                copy /= ten;
                Some(res)
            }
        })
    }
}
pub trait Square {
    fn square(self) -> Self;
}
impl<T: Mul<Output = T> + Copy> Square for T {
    fn square(self) -> Self {
        self * self
    }
}
}
pub mod primes {
pub mod prime {
use crate::algo_lib::misc::random::random;
use crate::algo_lib::numbers::gcd::gcd;
use crate::algo_lib::numbers::mod_int::ModInt;
use crate::algo_lib::numbers::num_traits::algebra::{One, Zero};
use crate::algo_lib::numbers::num_traits::primitive::Primitive;
use crate::algo_lib::numbers::number_ext::Power;
use crate::{dynamic_value, when};
pub fn is_prime(n: impl Primitive<i64>) -> bool {
    let n = n.to();
    if n <= 1 {
        return false;
    }
    let mut s = 0;
    let mut d = n - 1;
    while d % 2 == 0 {
        s += 1;
        d >>= 1;
    }
    if s == 0 {
        return n == 2;
    }
    dynamic_value!(IsPrimeModule : i64 = n);
    type Mod = ModInt<i64, IsPrimeModule>;
    for _ in 0..20 {
        let a = Mod::new(random().gen_bound(n as u64) as i64);
        if a == Mod::zero() {
            continue;
        }
        if a.power(d) == Mod::one() {
            continue;
        }
        let mut dd = d;
        let mut good = true;
        for _ in 0..s {
            if a.power(dd) == -Mod::one() {
                good = false;
                break;
            }
            dd *= 2;
        }
        if good {
            return false;
        }
    }
    true
}
pub fn next_prime(mut n: i64) -> i64 {
    if n <= 2 {
        return 2;
    }
    n += 1 - (n & 1);
    while !is_prime(n) {
        n += 2;
    }
    n
}
fn brent(n: i64, x0: i64, c: i64) -> i64 {
    dynamic_value!(ModVal : i64 = n);
    type Mod = ModInt<i64, ModVal>;
    let mut x = Mod::new(x0);
    let c = Mod::new(c);
    let mut g = 1;
    let mut q = Mod::one();
    let mut xs = Mod::zero();
    let mut y = Mod::zero();
    let m = 128i64;
    let mut l = 1;
    while g == 1 {
        y = x;
        for _ in 1..l {
            x = x * x + c;
        }
        let mut k = 0;
        while k < l && g == 1 {
            xs = x;
            for _ in 0..m.min(l - k) {
                x = x * x + c;
                q *= y - x;
            }
            g = gcd(q.val(), n);
            k += m;
        }
        l *= 2;
    }
    if g == n {
        loop {
            xs = xs * xs + c;
            g = gcd((xs - y).val(), n);
            if g != 1 {
                break;
            }
        }
    }
    g
}
pub fn find_divisor(n: i64) -> i64 {
    when! {
        n == 1 => 1, n % 2 == 0 => 2, is_prime(n) => n, else => { loop { let res =
        brent(n, random().gen_range(2..n), random().gen_range(1..n),); if res != n {
        return res; } } },
    }
}
}
}
}
}