// https://contest.ucup.ac/contest/1843/problem/9553
pub mod solution {
//{"name":"F. The Hermit","group":"Universal Cup - The 3rd Universal Cup. Stage 17: Jinan","url":"https://contest.ucup.ac/contest/1843/problem/9553","interactive":false,"timeLimit":1000,"tests":[{"input":"4 3\n","output":"7\n"},{"input":"11 4\n","output":"1187\n"},{"input":"100000 99999\n","output":"17356471\n"}],"testType":"single","input":{"type":"stdin","fileName":null,"pattern":null},"output":{"type":"stdout","fileName":null,"pattern":null},"languages":{"java":{"taskClass":"FTheHermit"}}}

use crate::algo_lib::io::input::Input;
use crate::algo_lib::io::output::Output;
use crate::algo_lib::misc::memo::memoization::Memoization2;
use crate::algo_lib::misc::recursive_function::Callable2;
use crate::algo_lib::misc::test_type::TaskType;

use crate::algo_lib::misc::test_type::TestType;
use crate::algo_lib::numbers::mod_int::ModIntF;
use crate::algo_lib::numbers::mod_utils::Combinations;
use crate::algo_lib::numbers::num_traits::algebra::One;
use crate::algo_lib::numbers::num_traits::algebra::Zero;
use crate::algo_lib::numbers::num_traits::as_index::AsIndex;

type PreCalc = ();

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

    type Mod = ModIntF;
    let c = Combinations::<Mod>::new(m + 1);
    let mut mem = Memoization2::new(|mem, n: usize, len: usize| -> (Mod, Mod) {
        if n <= len {
            (Mod::zero(), Mod::zero())
        } else if len == 0 {
            (Mod::zero(), Mod::one())
        } else {
            let mut qty = c.c(n - 1, len);
            let mut sum = Mod::zero();
            for i in 2..=n {
                let (call_sum, call_qty) = mem.call(n / i, len - 1);
                qty -= call_qty;
                sum += call_sum;
            }
            sum += qty * Mod::from_index(len);
            (sum, c.c(n - 1, len))
        }
    });
    out.print_line(mem.call(m, n).0 + mem.call(m, n - 1).0);
}

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

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

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

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

pub mod collections {
pub mod fx_hash_map {
// Copyright 2015 The Rust Project Developers. See the COPYRIGHT at
// http://rust-lang.org/COPYRIGHT.
//
// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
// <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your
// option. This file may not be copied, modified, or distributed
// except according to those terms.

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

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

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

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

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

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

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

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

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

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

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

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

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

    #[inline]
    fn finish(&self) -> u64 {
        self.hash as u64
    }
}
}
pub mod 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)
        }
    }

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

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

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

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

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

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

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

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

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

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

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;
use std::io::Stderr;
use std::io::Write;

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

static mut ERR: Option<Stderr> = None;

pub fn err() -> Output<'static> {
    unsafe {
        if ERR.is_none() {
            ERR = Some(stderr());
        }
        Output::new_with_auto_flush(ERR.as_mut().unwrap())
    }
}
}
}
pub mod misc {
pub mod memo {
pub mod memoization {
use crate::algo_lib::collections::fx_hash_map::FxHashMap;
use crate::algo_lib::misc::recursive_function::Callable;
use crate::algo_lib::misc::recursive_function::Callable2;
use crate::algo_lib::misc::recursive_function::Callable3;
use crate::algo_lib::misc::recursive_function::Callable4;
use crate::algo_lib::misc::recursive_function::Callable5;
use crate::algo_lib::misc::recursive_function::Callable6;
use crate::algo_lib::misc::recursive_function::Callable7;
use crate::algo_lib::misc::recursive_function::Callable8;
use crate::algo_lib::misc::recursive_function::Callable9;
use std::hash::Hash;

macro_rules! memoization {
    ($name: ident, $trait: ident, ($($type: ident $arg: ident,)*)) => {
        pub struct $name<F, $($type, )*Output>
        where
            F: FnMut(&mut dyn $trait<$($type, )*Output>, $($type, )*) -> Output,
        {
            f: std::cell::UnsafeCell<F>,
            res: FxHashMap<($($type, )*), Output>,
        }

        impl<F, $($type, )*Output: Clone> $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),
                    res: FxHashMap::default(),
                }
            }
        }

        impl<F, $($type : Hash + Eq + Clone, )*Output: Clone> $trait<$($type, )*Output> for $name<F, $($type, )*Output>
        where
            F: FnMut(&mut dyn $trait<$($type, )*Output>, $($type, )*) -> Output,
        {
            fn call(&mut self, $($arg: $type, )*) -> Output {
                match self.res.get(&($($arg.clone(), )*)).cloned() {
                    None => {
                        let res = unsafe { (*self.f.get())(self, $($arg.clone(), )*) };
                        self.res.insert(($($arg, )*), res.clone());
                        res
                    }
                    Some(res) => res,
                }
            }
        }
    }
}

memoization!(Memoization, Callable, (Arg arg,));
memoization!(Memoization2, Callable2, (Arg1 arg1, Arg2 arg2,));
memoization!(Memoization3, Callable3, (Arg1 arg1, Arg2 arg2, Arg3 arg3,));
memoization!(Memoization4, Callable4, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4,));
memoization!(Memoization5, Callable5, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4, Arg5 arg5,));
memoization!(Memoization6, Callable6, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4, Arg5 arg5, Arg6 arg6,));
memoization!(Memoization7, Callable7, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4, Arg5 arg5, Arg6 arg6, Arg7 arg7,));
memoization!(Memoization8, Callable8, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4, Arg5 arg5, Arg6 arg6, Arg7 arg7, Arg8 arg8,));
memoization!(Memoization9, Callable9, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4, Arg5 arg5, Arg6 arg6, Arg7 arg7, Arg8 arg8, Arg9 arg9,));
}
}
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 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> {
    //noinspection RsSelfConvention
    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);

        $name::set_val($val);
    };
    ($name: ident: $t: ty = $val: expr, $val_static: ident) => {
        dynamic_value!($name: $t, $val_static);

        $name::set_val($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;
use crate::algo_lib::numbers::num_traits::algebra::IntegerSemiRingWithSub;
use crate::algo_lib::numbers::num_traits::algebra::One;
use crate::algo_lib::numbers::num_traits::algebra::SemiRingWithSub;
use crate::algo_lib::numbers::num_traits::algebra::Zero;
use crate::algo_lib::numbers::num_traits::wideable::Wideable;
use std::mem::swap;

pub fn extended_gcd<T: IntegerSemiRingWithSub + Wideable + Copy>(a: T, b: T) -> (T, T::W, T::W)
where
    T::W: Copy + SemiRingWithSub,
{
    if a == T::zero() {
        (b, T::W::zero(), T::W::one())
    } else {
        let (d, y, mut x) = extended_gcd(b % a, a);
        x -= T::W::from(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 mod mod_int {
use crate::algo_lib::collections::fx_hash_map::FxHashMap;
use crate::algo_lib::io::input::Input;
use crate::algo_lib::io::input::Readable;
use crate::algo_lib::io::output::Output;
use crate::algo_lib::io::output::Writable;
use crate::algo_lib::misc::value::Value;
use crate::algo_lib::numbers::gcd::extended_gcd;
use crate::algo_lib::numbers::num_traits::algebra::Field;
use crate::algo_lib::numbers::num_traits::algebra::IntegerRing;
use crate::algo_lib::numbers::num_traits::algebra::One;
use crate::algo_lib::numbers::num_traits::algebra::Ring;
use crate::algo_lib::numbers::num_traits::algebra::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;
use crate::when;
use std::fmt::Display;
use std::fmt::Formatter;
use std::hash::Hash;
use std::marker::PhantomData;
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::Sub;
use std::ops::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;
}

#[derive(Copy, Clone, Eq, PartialEq, Hash, Default)]
pub struct ModInt<T, V: Value<T>> {
    n: T,
    phantom: PhantomData<V>,
}

impl<T: Copy, V: Value<T>> ModInt<T, V> {
    pub fn val(&self) -> T {
        self.n
    }
}

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())) }
    }
}

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(self.n, V::val());
        if g != T::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()
    }
}

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 mod_utils {
use crate::algo_lib::numbers::mod_int::BaseModInt;
use crate::algo_lib::numbers::num_traits::as_index::AsIndex;
use crate::algo_lib::numbers::num_utils::factorial;
use crate::algo_lib::numbers::num_utils::factorials;

pub fn inverses<M: BaseModInt>(len: usize) -> Vec<M>
where
    M::T: AsIndex,
{
    let mut res = Vec::new();
    if len > 0 {
        res.push(M::zero());
    }
    if len > 1 {
        res.push(M::one());
    }
    while res.len() < len {
        res.push(
            res[M::module().to_index() % res.len()]
                * (M::from(M::module() / (M::T::from_index(res.len()))).neg()),
        );
    }
    res
}

pub fn inverse_factorials<M: BaseModInt>(len: usize) -> Vec<M>
where
    M::T: AsIndex,
{
    let mut res = inverses(len);
    if len > 0 {
        res[0] = M::one();
    }
    for i in 1..len {
        let last = res[i - 1];
        res[i] *= last;
    }
    res
}

pub struct Combinations<M: BaseModInt>
where
    M::T: AsIndex,
{
    fact: Vec<M>,
    inv_fact: Vec<M>,
}

impl<M: BaseModInt + AsIndex> Combinations<M>
where
    M::T: AsIndex,
{
    pub fn new(len: usize) -> Self {
        Self {
            fact: factorials(len),
            inv_fact: inverse_factorials(len),
        }
    }

    pub fn c(&self, n: usize, k: usize) -> M {
        if n < k {
            M::zero()
        } else {
            self.fact[n] * self.inv_fact[k] * self.inv_fact[n - k]
        }
    }

    pub fn comb_with_rep(&self, n: usize, k: usize) -> M {
        self.c(n + k - 1, k)
    }

    pub fn c_inv(&self, n: usize, k: usize) -> M {
        self.inv_fact[n] * self.fact[k] * self.fact[n - k]
    }

    pub fn fact(&self, n: usize) -> M {
        self.fact[n]
    }

    pub fn inv_fact(&self, n: usize) -> M {
        self.inv_fact[n]
    }
}

pub fn combinations<M: BaseModInt + AsIndex>(n: usize, mut k: usize) -> M {
    if k > n {
        return M::zero();
    }
    if k > n - k {
        k = n - k;
    }
    let mut res = M::one();
    for i in n - k + 1..=n {
        res *= M::from_index(i);
    }
    res /= factorial(k);
    res
}
}
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 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 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;
use crate::algo_lib::numbers::num_traits::algebra::IntegerRing;
use crate::algo_lib::numbers::num_traits::algebra::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
    }
}
}
}
}
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);
}