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 C++ (GCC 9.2.1) (judge verdict: Accepted)

/**
 * code generated by JHelper
 * More info: https://github.com/AlexeyDmitriev/JHelper
 * @author
 */

#include <bits/stdc++.h>
using namespace std;
using ll = long long;

//#pragma GCC optimize("O3")
//#pragma GCC optimize("unroll-loops")

#define VI vector<int>
#define G(size_1) vector<vector<int>>(size_1, vector<int>())
#define SZ(x) ((int)(x).size())
#define READ ({int t;cin >> t;t;})
#define PII pair<int, int>

#define FOR(i, _begin, _end) for (__typeof(_end) end = _end, begin = _begin,  i = (begin) - ((begin) > (end)); i != (end) - ((begin) > (end)); i += 1 - 2 * ((begin) > (end)))
#define REP(i, end) for (__typeof(end) i = 0, _len = (end); i < (_len); i += 1)
#define ALL(x) (x).begin(),(x).end()
#define F first
#define S second
#define y0 y3487465
#define y1 y8687969
#define j0 j1347829
#define j1 j234892
#define MOD(x, m) ((((x) % (m)) + (m)) % (m))
#define BIT(n) (1LL<<(n))
#define UNIQUE(v) v.erase( unique(v.begin(), v.end()), v.end() );
#define EB emplace_back
#define PB push_back
#define fcout cout << fixed << setprecision(12)
#define fcerr cerr << fixed << setprecision(12)
#define print(x) cout << (x) << endl
#define fprint(x) cout << fixed << setprecision(12) << (x) << endl
# define BYE(a) do { cout << (a) << endl; return ; } while (false)

#ifdef DEBUG
#define ERR(args...) { string _s = #args; replace(_s.begin(), _s.end(), ',', ' '); stringstream _ss(_s); istream_iterator<string> _it(_ss); _err(cerr,_it, args); }
#define DBG(x) cerr << #x << " is " << x << endl;
#else
#define DBG(x) {};
#define ERR(args...) {};
#endif
void _err(std::ostream& cerr,istream_iterator<string> it) {cerr << endl;}
template<typename T, typename... Args>
void _err(std::ostream& cerr, istream_iterator<string> it, T a, Args... args) {
    cerr << *it << " = " << a << "  ";
    _err(cerr,++it, args...);
}

const double pi = 2 * acos(.0);
const int inf = 0x3f3f3f3f;
const ll mod = (ll) (1e9) + 7;

template<class T>bool Chmax(T &a, const T &b) { if (a<b) { a=b; return 1; } return 0; }
template<class T>bool Chmin(T &a, const T &b) { if (b<a) { a=b; return 1; } return 0; }

template <typename T>
istream& operator >> (istream& is, vector<T>& V) {
    for(auto&& ele : V)is >> ele;
    return is;
}

template <typename T>
ostream& operator << (ostream& os, const vector<T> V) {
    os << "[";
    int cnt = 0;
    T curr;
    if(!V.empty()){
        for (int i = 0; i < V.size() - 1; ++i) {
            if(V[i] == curr)cnt ++;
            else cnt = 0;
            if(cnt == 4)os << "... ";
            if(cnt < 4)
                os << i << ":" << V[i] << " ";
            curr = V[i];
        }
        os << V.size() - 1 << ":" << V.back();
    }
    os << "]\n";
    return os;
}

template <typename T, typename U>
ostream& operator << (ostream& os, const pair<T,U> P) {
    os << "(";
    os << P.first << "," << P.second;
    os << ")";
    return os;
}

template <typename T, typename U>
ostream& operator << (ostream& os, const set<T,U> V) {
    os << "{";
    if(!V.empty()){
        auto it = V.begin();
        for (int i = 0; i < V.size() -1; ++i) {
            os << *it << " ";
            it++;
        }
        os << *it;
    }
    os << "}\n";
    return os;
}

template <typename K, typename H, typename P>
ostream& operator << (ostream& os, const unordered_set<K, H, P> V) {
    os << "{";
    if(!V.empty()){
        auto it = V.begin();
        for (int i = 0; i < V.size() -1; ++i) {
            os << *it << " ";
            it++;
        }
        os << *it;
    }
    os << "}\n";
    return os;
}

template <typename K, typename C>
ostream& operator << (ostream& os, const multiset<K, C> V) {
    os << "{";
    if(!V.empty()){
        auto it = V.begin();
        for (int i = 0; i < V.size() -1; ++i) {
            os << *it << " ";
            it++;
        }
        os << *it;
    }
    os << "}";
    return os;
}

template <typename K, typename T, typename C>
ostream& operator << (ostream& os, const map<K,T,C> V) {
    os << "{";
    if(!V.empty()){
        auto it = V.begin();
        for (int i = 0; i < V.size() -1; ++i) {
            os << "(";
            os << it->first << "," << it->second;
            os << ") ";
            it++;
        }
        os << "(";
        os << it->first << "," << it->second;
        os << ")";
    }
    os << "}\n";
    return os;
}

template <typename K, typename T, typename C>
ostream& operator << (ostream& os, const unordered_map<K,T,C> V) {
    os << "{";
    if(!V.empty()){
        auto it = V.begin();
        for (int i = 0; i < V.size() -1; ++i) {
            os << "(";
            os << it->first << "," << it->second;
            os << ") ";
            it++;
        }
        os << "(";
        os << it->first << "," << it->second;
        os << ")";
    }
    os << "}\n";
    return os;
}

template <typename T>
ostream& operator << (ostream& os, const deque<T> V) {
    os << "[";
    if (!V.empty()) {
        for (int i = 0; i < V.size() - 1; ++i) {
            os << V[i] << "->";
        }
        if (!V.empty())os << V.back();
    }
    os << "]\n";
    return os;
};

template <typename T, typename Cont, typename Comp>
ostream& operator << (ostream& os, const priority_queue<T, Cont, Comp> V) {
    priority_queue<T, Cont, Comp> _V = V;
    os << "[";
    if(!_V.empty()){
        while(_V.size() > 1){
            os << _V.top() << "->";
            _V.pop();
        }
        os << _V.top();
    }
    os << "]\n";
    return os;
};

template <class F>
struct y_combinator {
    F f; // the lambda will be stored here

    // a forwarding operator():
    template <class... Args>
    decltype(auto) operator()(Args&&... args) const {
        // we pass ourselves to f, then the arguments.
        // the lambda should take the first argument as `auto&& recurse` or similar.
        return f(*this, std::forward<Args>(args)...);
    }
};
// helper function that deduces the type of the lambda:
template <class F>
y_combinator<std::decay_t<F>> recursive(F&& f){
return {std::forward<F>(f)};
}

struct hash_pair {
    template <class T1, class T2>
    size_t operator()(const pair<T1, T2>& p) const
    {
        auto hash1 = hash<T1>{}(p.first);
        auto hash2 = hash<T2>{}(p.second);
        return hash1 ^ hash2;
    }

};

template <typename T, typename U>
std::vector<T> multi_vector(int n, U v) {
    return std::vector<T>(n, v);
}

template <typename U, typename... Args>
auto multi_vector(int n, Args... args) {
    auto val = multi_vector<U>(std::forward<Args>(args)...);
    return std::vector<decltype(val)>(n, std::move(val));
}
template <signed M, unsigned T>
struct mod_int {
    constexpr static signed MODULO = M;
    constexpr static unsigned TABLE_SIZE = T;

    signed x;

    mod_int() : x(0) {}

    mod_int(long long y) : x(static_cast<signed>(y >= 0 ? y % MODULO : MODULO - (-y) % MODULO)) {}

    mod_int(int y) : x(y >= 0 ? y % MODULO : MODULO - (-y) % MODULO) {}


    explicit operator int() const
    {
        return x;
    }

    explicit operator long long() const
    {
        return x;
    }

    explicit operator double() const
    {
        return x;
    }

    mod_int &operator+=(const mod_int &rhs) {
        if ((x += rhs.x) >= MODULO) x -= MODULO;
        return *this;
    }

    mod_int &operator-=(const mod_int &rhs) {
        if ((x += MODULO - rhs.x) >= MODULO) x -= MODULO;
        return *this;
    }

    mod_int &operator*=(const mod_int &rhs) {
        x = static_cast<signed>(1LL * x * rhs.x % MODULO);
        return *this;
    }

    mod_int &operator/=(const mod_int &rhs) {
        x = static_cast<signed>((1LL * x * rhs.inv().x) % MODULO);
        return *this;
    }

    mod_int operator-() const { return mod_int(-x); }

    mod_int operator+(const mod_int &rhs) const { return mod_int(*this) += rhs; }

    mod_int operator-(const mod_int &rhs) const { return mod_int(*this) -= rhs; }

    mod_int operator*(const mod_int &rhs) const { return mod_int(*this) *= rhs; }

    mod_int operator/(const mod_int &rhs) const { return mod_int(*this) /= rhs; }

    bool operator<(const mod_int &rhs) const { return x < rhs.x; }

    mod_int inv() const {
        assert(x != 0);
        if (x <= static_cast<signed>(TABLE_SIZE)) {
            if (_inv[1].x == 0) prepare();
            return _inv[x];
        } else {
            signed a = x, b = MODULO, u = 1, v = 0, t;
            while (b) {
                t = a / b;
                a -= t * b;
                std::swap(a, b);
                u -= t * v;
                std::swap(u, v);
            }
            return mod_int(u);
        }
    }

    mod_int pow(long long t) const {
        assert(!(x == 0 && t == 0));
        mod_int e = *this, res = mod_int(1);
        for (; t; e *= e, t >>= 1)
            if (t & 1) res *= e;
        return res;
    }

    mod_int fact() {
        if (_fact[0].x == 0) prepare();
        return _fact[x];
    }

    mod_int inv_fact() {
        if (_fact[0].x == 0) prepare();
        return _inv_fact[x];
    }

    mod_int choose(mod_int y) {
        assert(y.x <= x);
        return this->fact() * y.inv_fact() * mod_int(x - y.x).inv_fact();
    }

    static mod_int _inv[TABLE_SIZE + 1];

    static mod_int _fact[TABLE_SIZE + 1];

    static mod_int _inv_fact[TABLE_SIZE + 1];

    static void prepare() {
        _inv[1] = 1;
        for (int i = 2; i <= (int)TABLE_SIZE; ++i) {
            _inv[i] = 1LL * _inv[MODULO % i].x * (MODULO - MODULO / i) % MODULO;
        }
        _fact[0] = 1;
        for (unsigned i = 1; i <= TABLE_SIZE; ++i) {
            _fact[i] = _fact[i - 1] * int(i);
        }
        _inv_fact[TABLE_SIZE] = _fact[TABLE_SIZE].inv();
        for (int i = (int)TABLE_SIZE - 1; i >= 0; --i) {
            _inv_fact[i] = _inv_fact[i + 1] * (i + 1);
        }
    }
};

template <int M, unsigned F>
std::ostream &operator<<(std::ostream &os, const mod_int<M, F> &rhs) {
    return os << rhs.x;
}

template <int M, unsigned F>
std::istream &operator>>(std::istream &is, mod_int<M, F> &rhs) {
    long long s;
    is >> s;
    rhs = mod_int<M, F>(s);
    return is;
}

template <int M, unsigned F>
mod_int<M, F> mod_int<M, F>::_inv[TABLE_SIZE + 1];

template <int M, unsigned F>
mod_int<M, F> mod_int<M, F>::_fact[TABLE_SIZE + 1];

template <int M, unsigned F>
mod_int<M, F> mod_int<M, F>::_inv_fact[TABLE_SIZE + 1];

template <int M, unsigned F>
bool operator==(const mod_int<M, F> &lhs, const mod_int<M, F> &rhs) {
    return lhs.x == rhs.x;
}

template <int M, unsigned F>
bool operator!=(const mod_int<M, F> &lhs, const mod_int<M, F> &rhs) {
    return !(lhs == rhs);
}

const int MF = 1000010;
const int MOD = 998244353;

using mint = mod_int<MOD, MF>;

mint binom(int n, int r) { return (r < 0 || r > n || n < 0) ? 0 : mint(n).choose(r); }

mint fact(int n) { return mint(n).fact(); }

mint inv_fact(int n) { return mint(n).inv_fact(); }
class FPass {
public:

void solve(std::istream& cin, std::ostream& cout, std::ostream& cerr) {
    string s;cin >> s;
    int N = SZ(s);
    VI a(N);
    REP(i,N)a[i] = s[i] - '0';
    VI sum_l(2*N+1), sum_m(2*N+1);
    VI t_a = a;
    ERR(t_a)
    REP(j,2*N){
        if(t_a[0] > 0){t_a[0]--, sum_m[j+1] = sum_m[j] + 1;}
        else sum_m[j+1] = sum_m[j];
        FOR(i,1,N)if(t_a[i] > 0)t_a[i-1]++, t_a[i]--;
//        ERR(t_a)
    }
    REP(i,N)t_a[i] = 2 - a[i];
    REP(j,2*N){
        if(t_a[0] > 0){t_a[0]--, sum_l[j+1] = sum_l[j] + 1;}
        else sum_l[j+1] = sum_l[j];
        FOR(i,1,N)if(t_a[i] > 0)t_a[i-1]++, t_a[i]--;
    }
    REP(i,2*N+1)sum_l[i] =  i - sum_l[i];
    ERR(sum_l, sum_m)
    auto dp = multi_vector<mint>(2*N+1, 2*N+1, 0);
    dp[0][0] = 1;
    REP(i,2*N)FOR(n,sum_l[i+1],sum_m[i+1]+1){
        ERR(i,n)
        dp[i+1][n] += dp[i][n];
        if(n-1 >= sum_l[i])dp[i+1][n] += dp[i][n-1];
    }
    print(dp[2*N][sum_m[2*N]]);

}


};


#undef int
int main() {

	FPass solver;
	std::istream& in(std::cin);
	std::ostream& out(std::cout);
    std::ostringstream err;
	in.tie(0); ios::sync_with_stdio(0);
    solver.solve(in, out,err);
	return 0;
}