Problem:
Score : 900 points

Problem Statement

There are N Snukes lining up in a row.
You are given a string S of length N. The i-th Snuke from the front has two red balls if the i-th character in S is 0; one red ball and one blue ball if the i-th character in S is 1; two blue balls if the i-th character in S is 2.

Takahashi has a sequence that is initially empty. Find the number of the possible sequences he may have after repeating the following procedure 2N times, modulo 998244353:

Each Snuke who has one or more balls simultaneously chooses one of his balls and hand it to the Snuke in front of him, or hand it to Takahashi if he is the first Snuke in the row.

Takahashi receives the ball and put it to the end of his sequence.

Constraints

1 \leq |S| \leq 2000

S consists of 0,1 and 2.

Note that the integer N is not directly given in input; it is given indirectly as the length of the string S.

Input

Input is given from Standard Input in the following format:

S

Output

Print the number of the possible sequences Takahashi may have after repeating the procedure 2N times, modulo 998244353.

Sample Input 1

02

Sample Output 1

3

There are three sequences that Takahashi may have: rrbb, rbrb and rbbr, where r and b stand for red and blue balls, respectively.

Sample Input 2

1210

Sample Output 2

55

Sample Input 3

12001021211100201020

Sample Output 3

543589959

Solution in Rust (1.15.1) (judge verdict: Accepted)

#[allow(unused_imports)]
use std::cmp::*;
#[allow(unused_imports)]
use std::collections::*;
use std::io::{Write, BufWriter};
// https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8
macro_rules! input {
    ($($r:tt)*) => {
        let stdin = std::io::stdin();
        let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));
        let mut next = move || -> String{
            bytes
                .by_ref()
                .map(|r|r.unwrap() as char)
                .skip_while(|c|c.is_whitespace())
                .take_while(|c|!c.is_whitespace())
                .collect()
        };
        input_inner!{next, $($r)*}
    };
}

macro_rules! input_inner {
    ($next:expr) => {};
    ($next:expr, ) => {};

    ($next:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };
}

macro_rules! read_value {
    ($next:expr, ( $($t:tt),* )) => {
        ( $(read_value!($next, $t)),* )
    };

    ($next:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    };

    ($next:expr, chars) => {
        read_value!($next, String).chars().collect::<Vec<char>>()
    };

    ($next:expr, usize1) => {
        read_value!($next, usize) - 1
    };

    ($next:expr, [ $t:tt ]) => {{
        let len = read_value!($next, usize);
        (0..len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    }};

    ($next:expr, $t:ty) => {
        $next().parse::<$t>().expect("Parse error")
    };
}

/// Verified by https://atcoder.jp/contests/arc093/submissions/3968098
mod mod_int {
    use std::ops::*;
    pub trait Mod: Copy { fn m() -> i64; }
    #[derive(Copy, Clone, Hash, PartialEq, Eq, PartialOrd, Ord)]
    pub struct ModInt<M> { pub x: i64, phantom: ::std::marker::PhantomData<M> }
    impl<M: Mod> ModInt<M> {
        fn check_integrity(self) {
            debug_assert!(self.x >= 0);
            debug_assert!(self.x < M::m());
        }
        // x >= 0
        pub fn new(x: i64) -> Self { ModInt::new_internal(x % M::m()) }
        fn new_internal(x: i64) -> Self {
            ModInt { x: x, phantom: ::std::marker::PhantomData }
        }
        #[allow(dead_code)]
        pub fn mul_fast(self, other: Self) -> Self {
            self.check_integrity();
            other.check_integrity();
            ModInt::new_internal(self.x * other.x % M::m())
        }
        #[allow(dead_code)]
        pub fn mul_slow(self, other: Self) -> Self {
            // Naive multiplication in order to avoid overflow
            self.check_integrity();
            other.check_integrity();
            let mut sum = ModInt::new_internal(0);
            let mut cur = self;
            let mut e = other.x;
            if self.x < other.x {
                cur = other;
                e = self.x;
            }
            while e > 0 {
                if e % 2 == 1 { sum += cur; }
                cur += cur;
                e /= 2;
            }
            sum
        }
        pub fn pow(self, mut e: i64) -> Self {
            self.check_integrity();
            debug_assert!(e >= 0);
            let mut sum = ModInt::new_internal(1);
            let mut cur = self;
            while e > 0 {
                if e % 2 != 0 { sum *= cur; }
                cur *= cur;
                e /= 2;
            }
            sum
        }
        #[allow(dead_code)]
        pub fn inv(self) -> Self { self.pow(M::m() - 2) }
    }
    impl<M: Mod, T: Into<ModInt<M>>> Add<T> for ModInt<M> {
        type Output = Self;
        fn add(self, other: T) -> Self {
            let other = other.into();
            self.check_integrity();
            other.check_integrity();
            let mut sum = self.x + other.x;
            if sum >= M::m() { sum -= M::m(); }
            ModInt::new_internal(sum)
        }
    }
    impl<M: Mod, T: Into<ModInt<M>>> Sub<T> for ModInt<M> {
        type Output = Self;
        fn sub(self, other: T) -> Self {
            let other = other.into();
            self.check_integrity();
            other.check_integrity();
            let mut sum = self.x - other.x;
            if sum < 0 { sum += M::m(); }
            ModInt::new_internal(sum)
        }
    }
    impl<M: Mod, T: Into<ModInt<M>>> Mul<T> for ModInt<M> {
        type Output = Self;
        fn mul(self, other: T) -> Self { self.mul_fast(other.into()) }
    }
    impl<M: Mod, T: Into<ModInt<M>>> AddAssign<T> for ModInt<M> {
        fn add_assign(&mut self, other: T) { *self = *self + other; }
    }
    impl<M: Mod, T: Into<ModInt<M>>> SubAssign<T> for ModInt<M> {
        fn sub_assign(&mut self, other: T) { *self = *self - other; }
    }
    impl<M: Mod, T: Into<ModInt<M>>> MulAssign<T> for ModInt<M> {
        fn mul_assign(&mut self, other: T) { *self = *self * other; }
    }
    impl<M: Mod> Neg for ModInt<M> {
        type Output = Self;
        fn neg(self) -> Self { ModInt::new(0) - self }
    }
    impl<M> ::std::fmt::Display for ModInt<M> {
        fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
            self.x.fmt(f)
        }
    }
    impl<M> ::std::fmt::Debug for ModInt<M> {
        fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
            self.x.fmt(f)
        }
    }
    impl<M: Mod> From<i64> for ModInt<M> {
        fn from(x: i64) -> Self { Self::new(x) }
    }
} // mod mod_int

macro_rules! define_mod {
    ($struct_name: ident, $modulo: expr) => {
        #[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Hash)]
        struct $struct_name {}
        impl mod_int::Mod for $struct_name { fn m() -> i64 { $modulo } }
    }
}
const MOD: i64 = 998244353;
define_mod!(P, MOD);
type ModInt = mod_int::ModInt<P>;

fn solve() {
    let out = std::io::stdout();
    let mut out = BufWriter::new(out.lock());
    macro_rules! puts {
        ($($format:tt)*) => (write!(out,$($format)*).unwrap());
    }
    input! { s: chars }
    let n = s.len();
    let mut dp = vec![vec![ModInt::new(0); 2 * n + 1]; 2 * n + 1];
    dp[0][0] = ModInt::new(1);
    for i in 0..n {
        let (dx, dy) = match s[i] {
            '0' => (2, 0),
            '1' => (1, 1),
            '2' => (0, 2),
            _ => panic!(),
        };
        for j in 0..i + 1 {
            let k = i - j;
            dp[j + dx][k + dy] += dp[j][k];
            dp[j][k] = ModInt::new(0);
        }
        for j in 0..min(5, 2 * n + 1) {
            //eprintln!("dp'[{}] = {:?}", j, &dp[j][0..min(5, n + 1)]);
        }
        for j in 0..i + 3 {
            let k = i + 2 - j;
            if j > 0 {
                dp[j - 1][k] += dp[j][k];
            }
            if k > 0 {
                dp[j][k - 1] += dp[j][k];
            }
            dp[j][k] = ModInt::new(0);
        }
        for j in 0..min(5, 2 * n + 1) {
           // eprintln!("dp[{}] = {:?}", j, &dp[j][0..min(5, n + 1)]);
        }
    }
    for j in (0..2 * n + 1).rev() {
        for k in (0..2 * n + 1).rev() {
            if j > 0 {
                dp[j - 1][k] += dp[j][k];
            }
            if k > 0 {
                dp[j][k - 1] += dp[j][k];
            }
        }
    }
    let mut tot = ModInt::new(0);
    for i in 0..2 * n + 1 {
        for j in 0.. 2 * n + 1 {
            tot += dp[i][j];
        }
    }
    puts!("{}\n", dp[0][0]);
}

fn main() {
    // In order to avoid potential stack overflow, spawn a new thread.
    let stack_size = 104_857_600; // 100 MB
    let thd = std::thread::Builder::new().stack_size(stack_size);
    thd.spawn(|| solve()).unwrap().join().unwrap();
}