#line 2 "/home/user/competitive_programming/library_for_cpp/template/template.hpp"

using namespace std;

// intrinstic
#include <immintrin.h>

#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <cctype>
#include <cfenv>
#include <cfloat>
#include <chrono>
#include <cinttypes>
#include <climits>
#include <cmath>
#include <complex>
#include <cstdarg>
#include <cstddef>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <deque>
#include <fstream>
#include <functional>
#include <initializer_list>
#include <iomanip>
#include <ios>
#include <iostream>
#include <istream>
#include <iterator>
#include <limits>
#include <list>
#include <map>
#include <memory>
#include <new>
#include <numeric>
#include <ostream>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <streambuf>
#include <string>
#include <tuple>
#include <type_traits>
#include <typeinfo>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>

// utility
#line 2 "/home/user/competitive_programming/library_for_cpp/template/utility.hpp"

using ll = long long;

// a ← max(a, b) を実行する. a が更新されたら, 返り値が true.
template<typename T, typename U>
inline bool chmax(T &a, const U b){
    return (a < b ? a = b, 1: 0);
}

// a ← min(a, b) を実行する. a が更新されたら, 返り値が true.
template<typename T, typename U>
inline bool chmin(T &a, const U b){
    return (a > b ? a = b, 1: 0);
}

// a の最大値を取得する.
template<typename T>
inline T max(const vector<T> &a){
    if (a.empty()) throw invalid_argument("vector is empty.");

    return *max_element(a.begin(), a.end());
}

// vector<T> a の最小値を取得する.
template<typename T>
inline T min(const vector<T> &a){
    if (a.empty()) throw invalid_argument("vector is empty.");

    return *min_element(a.begin(), a.end());
}

// vector<T> a の最大値のインデックスを取得する.
template<typename T>
inline size_t argmax(const vector<T> &a){
    if (a.empty()) throw std::invalid_argument("vector is empty.");

    return distance(a.begin(), max_element(a.begin(), a.end()));
}

// vector<T> a の最小値のインデックスを取得する.
template<typename T>
inline size_t argmin(const vector<T> &a){
    if (a.empty()) throw invalid_argument("vector is empty.");

    return distance(a.begin(), min_element(a.begin(), a.end()));
}
#line 59 "/home/user/competitive_programming/library_for_cpp/template/template.hpp"

// math
#line 2 "/home/user/competitive_programming/library_for_cpp/template/math.hpp"

// 演算子
template<typename T>
T add(const T &x, const T &y) { return x + y; }

template<typename T>
T sub(const T &x, const T &y) { return x - y; }

template<typename T>
T mul(const T &x, const T &y) { return x * y; }

template<typename T>
T neg(const T &x) { return -x; }

template<typename T>
T bitwise_and(const T &x, const T &y) { return x & y; }

template<typename T>
T bitwise_or(const T &x, const T &y) { return x | y; }

template<typename T>
T bitwise_xor(const T &x, const T &y) { return x ^ y; }

// 除算に関する関数

// floor(x / y) を求める.
template<typename T, typename U>
T div_floor(T x, U y){ return (x > 0 ? x / y: (x - y + 1) / y); }

// ceil(x / y) を求める.
template<typename T, typename U>
T div_ceil(T x, U y){ return (x > 0 ? (x + y - 1) / y: x / y) ;}

// x を y で割った余りを求める.
template<typename T, typename U>
T safe_mod(T x, U y){
    T q = div_floor(x, y);
    return x - q * y ;
}

// x を y で割った商と余りを求める.
template<typename T, typename U>
pair<T, T> divmod(T x, U y){
    T q = div_floor(x, y);
    return {q, x - q * y};
}

// 四捨五入を求める.
template<typename T, typename U>
T round(T x, U y){
    T q, r;
    tie (q, r) = divmod(x, y);
    return (r >= div_ceil(y, 2)) ? q + 1 : q;
}

// 指数に関する関数

// x の y 乗を求める.
ll intpow(ll x, ll y){
    ll a = 1;
    while (y){
        if (y & 1) { a *= x; }
        x *= x;
        y >>= 1;
    }
    return a;
}

// x の y 乗を z で割った余りを求める.
ll modpow(ll x, ll y, ll z){
    ll a = 1;
    while (y){
        if (y & 1) { (a *= x) %= z; }
        (x *= x) %= z;
        y >>= 1;
    }
    return a;
}

// x の y 乗を z で割った余りを求める.
template<typename T, typename U>
T modpow(T x, U y, T z) {
    T a = 1;
    while (y) {
        if (y & 1) { (a *= x) %= z; }

        (x *= x) %= z;
        y >>= 1;
    }

    return a;
}

// vector の要素の総和を求める.
ll sum(vector<ll> &X){
    ll y = 0;
    for (auto &&x: X) { y+=x; }
    return y;
}

// vector の要素の総和を求める.
template<typename T>
T sum(vector<T> &X){
    T y = T(0);
    for (auto &&x: X) { y += x; }
    return y;
}

// a x + b y = gcd(a, b) を満たす整数の組 (a, b) に対して, (x, y, gcd(a, b)) を求める.
tuple<ll, ll, ll> Extended_Euclid(ll a, ll b) {
    ll s = 1, t = 0, u = 0, v = 1;
    while (b) {
        ll q;
        tie(q, a, b) = make_tuple(div_floor(a, b), b, safe_mod(a, b));
        tie(s, t) = make_pair(t, s - q * t);
        tie(u, v) = make_pair(v, u - q * v);
    }

    return make_tuple(s, u, a);
}

// floor(sqrt(N)) を求める (N < 0 のときは, 0 とする).
ll isqrt(const ll &N) { 
    if (N <= 0) { return 0; }

    ll x = sqrt(N);
    while ((x + 1) * (x + 1) <= N) { x++; }
    while (x * x > N) { x--; }

    return x;
}

// floor(sqrt(N)) を求める (N < 0 のときは, 0 とする).
ll floor_sqrt(const ll &N) { return isqrt(N); }

// ceil(sqrt(N)) を求める (N < 0 のときは, 0 とする).
ll ceil_sqrt(const ll &N) {
    ll x = isqrt(N);
    return x * x == N ? x : x + 1;
}
#line 62 "/home/user/competitive_programming/library_for_cpp/template/template.hpp"

// inout
#line 1 "/home/user/competitive_programming/library_for_cpp/template/inout.hpp"
// 入出力
template<class... T>
void input(T&... a){ (cin >> ... >> a); }

void print(){ cout << "\n"; }

template<class T, class... Ts>
void print(const T& a, const Ts&... b){
    cout << a;
    (cout << ... << (cout << " ", b));
    cout << "\n";
}

template<typename T, typename U>
istream &operator>>(istream &is, pair<T, U> &P){
    is >> P.first >> P.second;
    return is;
}

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

template<typename T>
vector<T> vector_input(int N, int index){
    vector<T> X(N+index);
    for (int i=index; i<index+N; i++) cin >> X[i];
    return X;
}

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

template<typename T>
ostream &operator<<(ostream &os, const vector<T> &X){
    int s = (int)X.size();
    for (int i = 0; i < s; i++) { os << (i ? " " : "") << X[i]; }
    return os;
}

template<typename T>
ostream &operator<<(ostream &os, const unordered_set<T> &S){
    int i = 0;
    for (T a: S) {os << (i ? " ": "") << a; i++;}
    return os;
}

template<typename T>
ostream &operator<<(ostream &os, const set<T> &S){
    int i = 0;
    for (T a: S) { os << (i ? " ": "") << a; i++; }
    return os;
}

template<typename T>
ostream &operator<<(ostream &os, const unordered_multiset<T> &S){
    int i = 0;
    for (T a: S) { os << (i ? " ": "") << a; i++; }
    return os;
}

template<typename T>
ostream &operator<<(ostream &os, const multiset<T> &S){
    int i = 0;
    for (T a: S) { os << (i ? " ": "") << a; i++; }
    return os;
}

template<typename T>
std::vector<T> input_vector(size_t n, size_t offset = 0) {
    std::vector<T> res;
    // 最初に必要な全容量を確保（再確保を防ぐ）
    res.reserve(n + offset);
    // offset 分をデフォルト値で埋める（特別 indexed 用）
    res.assign(offset, T());
    
    for (size_t i = 0; i < n; ++i) {
        T el;
        if (!(std::cin >> el)) break;
        res.push_back(std::move(el));
    }
    return res;
}
#line 65 "/home/user/competitive_programming/library_for_cpp/template/template.hpp"

// macro
#line 2 "/home/user/competitive_programming/library_for_cpp/template/macro.hpp"

// マクロの定義
#define all(x) x.begin(), x.end()
#define len(x) ll(x.size())
#define elif else if
#define unless(cond) if (!(cond))
#define until(cond) while (!(cond))
#define loop while (true)

// オーバーロードマクロ
#define overload2(_1, _2, name, ...) name
#define overload3(_1, _2, _3, name, ...) name
#define overload4(_1, _2, _3, _4, name, ...) name
#define overload5(_1, _2, _3, _4, _5, name, ...) name

// 繰り返し系
#define rep1(n) for (ll i = 0; i < n; i++)
#define rep2(i, n) for (ll i = 0; i < n; i++)
#define rep3(i, a, b) for (ll i = a; i < b; i++)
#define rep4(i, a, b, c) for (ll i = a; i < b; i += c)
#define rep(...) overload4(__VA_ARGS__, rep4, rep3, rep2, rep1)(__VA_ARGS__)

#define foreach1(x, a) for (auto &&x: a)
#define foreach2(x, y, a) for (auto &&[x, y]: a)
#define foreach3(x, y, z, a) for (auto &&[x, y, z]: a)
#define foreach4(x, y, z, w, a) for (auto &&[x, y, z, w]: a)
#define foreach(...) overload5(__VA_ARGS__, foreach4, foreach3, foreach2, foreach1)(__VA_ARGS__)
#line 68 "/home/user/competitive_programming/library_for_cpp/template/template.hpp"

// bitop
#line 2 "/home/user/competitive_programming/library_for_cpp/template/bitop.hpp"

// 非負整数 x の bit legnth を求める.
ll bit_length(ll x) {
    if (x == 0) { return 0; }
    return (sizeof(long) * CHAR_BIT) - __builtin_clzll(x);
}

// 非負整数 x の popcount を求める.
ll popcount(ll x) { return __builtin_popcountll(x); }

// 正の整数 x に対して, floor(log2(x)) を求める.
ll floor_log2(ll x) { return bit_length(x) - 1; }

// 正の整数 x に対して, ceil(log2(x)) を求める.
ll ceil_log2(ll x) { return bit_length(x - 1); }

// x の第 k ビットを取得する
int get_bit(ll x, int k) { return (x >> k) & 1; }

// x のビット列を取得する.
// k はビット列の長さとする.
vector<int> get_bits(ll x, int k) {
    vector<int> bits(k);
    rep(i, k) {
        bits[i] = x & 1;
        x >>= 1;
    }

    return bits;
}

// x のビット列を取得する.
vector<int> get_bits(ll x) { return get_bits(x, bit_length(x)); }
#line 71 "/home/user/competitive_programming/library_for_cpp/template/template.hpp"

// exception
#line 2 "/home/user/competitive_programming/library_for_cpp/template/exception.hpp"

class NotExist: public exception {
    private:
    string message;

    public:
    NotExist() : message("求めようとしていたものは存在しません.") {}

    const char* what() const noexcept override {
        return message.c_str();
    }
};
#line 2 "/home/user/competitive_programming/library_for_cpp/Algebra/modint.hpp"

#line 4 "/home/user/competitive_programming/library_for_cpp/Algebra/modint.hpp"

template<int M>
class modint {
    public:
    static constexpr int _mod = M; 
    uint64_t x;

    public:
    static int mod() { return _mod; }

    static modint raw(int v) {
        modint a;
        a.x = v;
        return a;
    }

    // 初期化
    constexpr modint(): x(0) {}
    constexpr modint(int64_t a) {
        int64_t w = (int64_t)(a) % mod();
        if (w < 0) { w += mod(); }
        x = w;
    }

    // マイナス元
    modint operator-() const { return modint(-x); }

    // 加法
    modint& operator+=(const modint &b){
        if ((x += b.x) >= mod()) x -= mod();
        return *this;
    }

    friend modint operator+(const modint &x, const modint &y) { return modint(x) += y; }

    // 減法
    modint& operator-=(const modint &b){
        if ((x += mod() - b.x) >= mod()) x -= mod();
        return *this;
    }

    friend modint operator-(const modint &x, const modint &y) { return modint(x) -= y; }

    // 乗法
    modint& operator*=(const modint &b){
        (x *= b.x) %= mod();
        return *this;
    }

    friend modint operator*(const modint &x, const modint &y) { return modint(x) *= y; }
    friend modint operator*(const int &x, const modint &y) { return modint(x) *= y; }
    friend modint operator*(const ll &x, const modint &y) { return modint(x) *= y; }

    // 除法
    modint& operator/=(const modint &b){ return (*this) *= b.inverse(); }

    friend modint operator/(const modint &x, const modint &y) { return modint(x) /= y; }

    modint inverse() const {
        int64_t s = 1, t = 0;
        int64_t a = x, b = mod();

        while (b > 0) {
            int64_t q = a / b;

            a -= q * b; swap(a, b);
            s -= q * t; swap(s, t);
        }

        assert (a == 1);

        return modint(s);
    }

    // 比較
    friend bool operator==(const modint &a, const modint &b) { return (a.x == b.x); }
    friend bool operator==(const modint &a, const int &b) { return a.x == safe_mod(b, mod()); }
    friend bool operator!=(const modint &a, const modint &b) { return (a.x != b.x); }

    // 入力
    friend istream &operator>>(istream &is, modint &a) {
        int64_t x;
        is >> x;
        a.x = safe_mod(x, mod());
        return is;
    }

    // 出力
    friend ostream &operator<<(ostream &os, const modint &a) { return os << a.x; }

    bool is_zero() const { return x == 0; }
    bool is_member(ll a) const { return x == (a % mod() + mod()) % mod(); }
};

template<int mod>
modint<mod> pow(modint<mod> x, long long n) {
    if (n < 0) { return pow(x, -n).inverse(); }

    auto res = modint<mod>(1);
    for (; n; n >>= 1) {
        if (n & 1) { res *= x; }
        x *= x;
    }

    return res;
}
#line 2 "/home/user/competitive_programming/library_for_cpp/Segment_Tree/Lazy_Segment_Tree.hpp"

/* 遅延セグメント木
M を Monoid とする. M 上の列に対して, Monid F からの区間作用と, 連続部分列に対する区間積の計算の処理を高速に行う.

* M: Monoid
* F: Monoid
* op: M x M → M: M 上の演算
* unit: M の単位元
* act: F x M → M: F からの M の演算
* comp: F x F → F: F 同士の合成 (左の要素が新しい)
* id: F の単位元

(条件)
M: Monoid, F = {f: F x M → M: 作用素} に対して, 以下が成立する.
* F は写像の合成に閉じている. つまり, 任意の f,g in F に対して, comp(f,g) in F
* F は M に作用する. つまり, 以下が成り立つ.
    * F の単位元 id は恒等的に作用する. つまり, 任意の x in M に対して id(x) = x となる.
    * 任意の f in F, x,y in M に対して, f(xy) = f(x) f(y) である.


(注意)
作用素は左から掛ける. 更新も左から行う.
*/

#line 27 "/home/user/competitive_programming/library_for_cpp/Segment_Tree/Lazy_Segment_Tree.hpp"

template<typename M, typename F>
class Lazy_Segment_Tree {
    public:
    int n, depth;
    const function<M(M, M)> op;
    const function<M(F, M)> act;
    const function<F(F, F)> comp;
    vector<M> data; const M unit;
    vector<F> lazy; const F id;

    public:
    Lazy_Segment_Tree(int size, const function<M(M, M)> op, const M unit, const function<M(F, M)> act, const function<F(F, F)> comp, const F id):
        n(), op(op), unit(unit), act(act), comp(comp), id(id), depth(0) {
            int m = 1;
            while (size > m) { depth++, m *= 2; }
            n = m;
            data.assign(2 * m, unit);
            lazy.assign(2 * m, id);
        }

    Lazy_Segment_Tree(const vector<M> &vec, const function<M(M, M)> op, const M unit, const function<M(F, M)> act, const function<F(F, F)> comp, const F id):
        Lazy_Segment_Tree(vec.size(), op, unit, act, comp, id){
            for (int k = 0; k < vec.size(); k++) { data[k+n] = vec[k]; }
            for (int k = n - 1; k > 0; k--) { data[k] = op(data[k << 1], data[k << 1 | 1]); }
        }

    private:
    inline M evaluate_at(int m){ return lazy[m] == id ? data[m] : act(lazy[m], data[m]); }

    /// @brief セグメントツリーの第 m 要素を更新し, 遅延していた作用を子に伝搬させる.
    /// @param m 
    void push(int m){
        data[m] = evaluate_at(m);

        if ((m < n) && (lazy[m] != id)){
            int left = m << 1;
            lazy[left] = (lazy[left] == id) ? lazy[m] : comp(lazy[m], lazy[left]);

            int right = m << 1 | 1;
            lazy[right] = (lazy[right] == id) ? lazy[m] : comp(lazy[m], lazy[right]); 
        }

        lazy[m] = id;
    }

    /// @brief セグメントツリーの第 m 要素を含む区間についての lazy の要素について, 子への更新を行う.
    /// @param m 
    inline void propagate_above(int m){
        int h = 0, mm = m;
        for (mm; mm; mm >>= 1, h++){}

        for (h--; h >= 0; h--) { push(m >> h); }
    }

    /// @brief セグメントツリーの第 m 要素を含む区間についての data の要素を更新する.
    /// @param m 
    inline void recalc_above(int m){
        while (m > 1){
            m >>= 1;
            data[m] = op(evaluate_at(m << 1), evaluate_at(m << 1 | 1));
        }
    }

    pair<int, int> range_propagate(int l, int r){
        int X = l + n, Y = r + n - 1, L0 = -1, R0 = -1;
    
        while (X < Y){
            if (X & 1) { L0 = max(L0, X++); }
            if ((Y & 1) ==0 ) { R0 = max(R0, Y--); }

            X >>= 1; Y >>= 1;
        }

        L0 = max(L0, X); R0 = max(R0, Y);
        propagate_above(L0); propagate_above(R0);
        return make_pair(L0, R0);
    }

    public:
    /// @brief 第 k 項を取得する.
    /// @param k 
    /// @return 第 k 項
    inline M operator[](int k){
        int m = k + n;
        propagate_above(m);
        lazy[m] = id;
        return data[m] = evaluate_at(m);
    }

    /// @brief i = l, l + 1, ..., r に対して, 第 i 項に対して alpha を作用させる.
    /// @param l 区間の左端
    /// @param r 区間の右端
    /// @param alpha 作用
    void action(int l, int r, F alpha){
        int L0, R0;
        tie(L0, R0) = range_propagate(l, r + 1);

        int L = l + n, R = r + n + 1;
        while (L < R){
            if (L & 1){
                lazy[L] = (lazy[L] == id) ? alpha : comp(alpha, lazy[L]); 
                L++;
            }

            if (R & 1){
                R--;
                lazy[R] = (lazy[R] == id) ? alpha : comp(alpha, lazy[R]);
            }

            L >>= 1; R >>= 1;
        }

        recalc_above(L0); recalc_above(R0);
    }

    /// @brief 第 k 項を x に更新する.
    /// @param k 更新場所
    /// @param x 更新後の要素
    inline void update(int k, M x){
        int m = k + n;
        propagate_above(m);
        data[m] = x; lazy[m] = id;
        recalc_above(m);
    }


    /// @brief 積 x[l] * x[l + 1] * ... * x[r] を求める.
    /// @param l 区間の左端
    /// @param r 区間の右端
    /// @return 積
    M product(int l, int r){
        int L0, R0;
        tie(L0, R0) = range_propagate(l, r + 1);

        int L = l + n, R = r + n + 1;
        M vL = unit, vR = unit;
        while (L < R){
            if (L & 1) { vL = op(vL, evaluate_at(L)); L++; }
            if (R & 1) { R--; vR=op(evaluate_at(R), vR); }

            L >>= 1; R >>= 1;
        }

        return op(vL, vR);
    }

    /// @brief 全要素における区間積を求める.
    /// @return 残要素における区間積
    inline M all_product() {return product(0, n - 1);}

    template<typename Func>
    int max_right(int l, const Func &cond) {
        assert(cond(unit));
        if (l == n) return n;

        l += n;
        propagate_above(l);
        M sm = unit;
        do {
            while (l % 2 == 0) l >>= 1;

            if (!cond(op(sm, evaluate_at(l)))) {
                while (l < n) {
                    push(l);
                    l <<= 1;
                    if (cond(op(sm, evaluate_at(l)))) {
                        sm = op(sm, evaluate_at(l));
                        l++;
                    }
                }
                return l - n;
            }
            sm = op(sm, evaluate_at(l));
            l++;
        } while ((l & -l) != l);
        return n;
    }

    void refresh() {
        for (int m = 1; m < 2 * n; m++){
            data[m] = evaluate_at(m);
            if ((m < n) && (lazy[m] != id)){
                int left = m << 1;
                lazy[left] = (lazy[left] == id) ? lazy[m] : comp(lazy[m], lazy[left]);

                int right = m << 1 | 1;
                lazy[right] = (lazy[right] == id) ? lazy[m] : comp(lazy[m], lazy[m << 1 | 1]);
            }
            lazy[m] = id;
        }
    }
};
#line 2 "/home/user/competitive_programming/library_for_cpp/Segment_Tree/preset/Range_Add_Range_Sum.hpp"

#line 4 "/home/user/competitive_programming/library_for_cpp/Segment_Tree/preset/Range_Add_Range_Sum.hpp"

template<typename T>
class Range_Add_Range_Sum_Lazy_Segment_Tree : public Lazy_Segment_Tree<pair<T, int>, T> {
    using M = pair<T, int>;
    using F = T;

    static M op(M x, M y) { return {x.first + y.first, x.second + y.second}; }
    static M act(F a, M x) { return {x.first + a * T(x.second), x.second}; }
    static F comp(F a, F b) { return a + b; }

    static vector<M> convert(const vector<T> &vec) {
        vector<M> res(vec.size());
        for (int i = 0; i < (int)vec.size(); ++i) {
            res[i] = {vec[i], 1};
        }
        return res;
    }

    public:
    Range_Add_Range_Sum_Lazy_Segment_Tree(int n) : Lazy_Segment_Tree<M, F>(
        vector<M>(n, {0, 1}), op, {0, 0}, act, comp, 0
    ) {}

    Range_Add_Range_Sum_Lazy_Segment_Tree(const vector<T> &vec) : Lazy_Segment_Tree<M, F>(
        convert(vec), op, {0, 0}, act, comp, 0
    ) {}

    void update(int k, T x) { Lazy_Segment_Tree<M, F>::update(k, {x, 1}); }

    T operator[](int k) { return Lazy_Segment_Tree<M, F>::operator[](k).first; }

    void add(int l, int r, T x) { this->action(l, r, x); }

    T all_sum() { return this->all_product().first; }
    T sum(int l, int r) { return this->product(l, r).first; }
};
#line 2 "/home/user/competitive_programming/library_for_cpp/Segment_Tree/preset/Range_Add_Range_Min.hpp"

#line 4 "/home/user/competitive_programming/library_for_cpp/Segment_Tree/preset/Range_Add_Range_Min.hpp"

template<typename T>
class Range_Add_Range_Min_Lazy_Segment_Tree : public Lazy_Segment_Tree<T, T> {
    using M = T;
    using F = T;

    static M op(M x, M y) { return x < y ? x : y; }
    static M act(F a, M x) { return x + a; }
    static F comp(F a, F b) { return a + b; }

    public:
    Range_Add_Range_Min_Lazy_Segment_Tree(int n, T first, T unit) : Lazy_Segment_Tree<M, F>(
        vector<M>(n, first), op, unit, act, comp, 0
    ) {}

    Range_Add_Range_Min_Lazy_Segment_Tree(int n, T unit) : Range_Add_Range_Min_Lazy_Segment_Tree<T>(n, unit, unit) {}

    Range_Add_Range_Min_Lazy_Segment_Tree(const vector<T> &vec, T unit) : Lazy_Segment_Tree<M, F>(
        vec, op, unit, act, comp, 0
    ) {}

    void update(int k, T x) { Lazy_Segment_Tree<M, F>::update(k, x); }

    T operator[](int k) { return Lazy_Segment_Tree<M, F>::operator[](k); }

    void add(int l, int r, T x) { this->action(l, r, x); }

    T min(int l, int r) { return this->product(l, r); }
};
#line 2 "/home/user/competitive_programming/library_for_cpp/Modulo_Polynomial/Numeric_Theory_Translation.hpp"

#line 2 "/home/user/competitive_programming/library_for_cpp/Modulo_Polynomial/Modulo_Polynomial.hpp"

#line 5 "/home/user/competitive_programming/library_for_cpp/Modulo_Polynomial/Modulo_Polynomial.hpp"

template<typename mint>
class Modulo_Polynomial {
    public:
    int precision = 0;

    public:
    vector<mint> poly;
    Modulo_Polynomial(vector<mint> _poly, int precision): precision(precision) {
        if (_poly.size() > precision) { _poly.resize(precision); }
        poly = _poly;
    }

    Modulo_Polynomial() = default;
    Modulo_Polynomial(vector<mint> poly) : Modulo_Polynomial(poly, poly.size()) {}
    Modulo_Polynomial(int precision) : Modulo_Polynomial({}, precision) {}

    // 演算子の定義
    public:
    // マイナス元
    Modulo_Polynomial operator-() const {
        Modulo_Polynomial res(*this);
        for (auto &a : res.poly) { a = -a; }
        return res;
    }

    // 加法
    Modulo_Polynomial& operator+=(const Modulo_Polynomial &P){
        if (size() < P.size()) { resize(P.size()); }

        for (int i = 0; i < (int) P.poly.size(); i++) { poly[i] += P[i]; }
        reduce();

        return *this;
    }

    Modulo_Polynomial& operator+=(const mint &a){
        if (poly.empty()) { resize(1); }
        poly[0] += a;
        reduce();

        return *this;
    }

    friend Modulo_Polynomial operator+(const Modulo_Polynomial &lhs, const Modulo_Polynomial &rhs) { return Modulo_Polynomial(lhs) += rhs; }
    Modulo_Polynomial operator+(const mint &a) const { return Modulo_Polynomial(*this) += a; }

    // 減法
    Modulo_Polynomial& operator-=(const Modulo_Polynomial &P){
        if (size() < P.size()) { resize(P.size()); }

        for (int i = 0; i < (int) P.poly.size(); i++) { poly[i] -= P[i]; }
        reduce();

        return *this;
    }

    Modulo_Polynomial& operator-=(const mint &a){
        if (poly.empty()) { resize(1); }
        poly[0] -= a;
        reduce();

        return *this;
    }

    friend Modulo_Polynomial operator-(const Modulo_Polynomial &lhs, const Modulo_Polynomial &rhs) { return Modulo_Polynomial(lhs) -= rhs; }
    Modulo_Polynomial operator-(const mint &a) const { return Modulo_Polynomial(*this) -= a; }

    // スカラー倍
    Modulo_Polynomial& operator*=(const mint &a){
        for (int i = 0; i < size(); i++) { poly[i] *= a; }
        reduce();
        return *this;
    }

    Modulo_Polynomial operator*(const mint &a) const {return Modulo_Polynomial(*this) *= a;}

    friend Modulo_Polynomial operator*(const mint &a, const Modulo_Polynomial &P) {
        Modulo_Polynomial res(P);
        res *= a;
        return res;
    }

    // 積
    Modulo_Polynomial& operator*=(const Modulo_Polynomial &P) {
        int r = min({(int) (poly.size() + P.poly.size()) - 1, precision, P.precision});
        vector<mint> A(r);
        for (int i = 0; i < size(); i++) {
            for (int j = 0; j < P.size(); j++) {
                if (i + j < r) { A[i + j] += poly[i] * P.poly[j]; }
            }
        }

        poly = A;
        precision = min(precision, P.precision);
        return *this;
    }

    friend Modulo_Polynomial operator*(const Modulo_Polynomial &lhs, const Modulo_Polynomial &rhs) { return Modulo_Polynomial(lhs) *= rhs; }

    // スカラー除算
    Modulo_Polynomial& operator/=(const mint &a) {
        mint a_inv = a.inverse();
        for (int i = 0; i < size(); i++) { poly[i] *= a_inv; }
        return *this;
    }

    Modulo_Polynomial operator/(const mint &a) const { return Modulo_Polynomial(*this) /= a; }

    // index
    mint operator[] (int k) const { return (k < poly.size()) ? poly[k] : 0; }

    // istream
    friend istream &operator>>(istream &is, Modulo_Polynomial &P) {
        P.poly.resize(P.precision);
        for (int i = 0; i < (int)P.precision; i++) { is >> P.poly[i]; }
        return (is);
    }

    // ostream
    friend ostream &operator<<(ostream &os, const Modulo_Polynomial &P){
        for (int i = 0; i < (int)P.poly.size(); i++){
            os << (i ? " " : "") << P[i];
        }
        return os;
    }

    // poly で保持しているベクトルの長さを size にする.
    // size = -1 のときは, size = precision に変換される.
    void resize(int size = -1) {
        if (size == -1) { size = this -> precision; }
        size = min(size, this -> precision);
        poly.resize(size);
    }

    bool is_zero() const {
        for (auto &a: poly) { unless(a.is_zero()) {return false;} }
        return true;
    }

    // 高次に連なる 0 を削除する
    void reduce() {
        while (!poly.empty() && poly.back().is_zero()) { poly.pop_back(); }
    }

    // 保持している多項式の乗法の項の長さを求める
    int size() const { return poly.size(); }

    // 次数を求める (ゼロ多項式の時は -1)
    int degree() const {
        for (int d = size() - 1; d >= 0; d--) {
            unless(poly[d].is_zero()) { return d; }
        }
        return -1;
    }

    // 位数 (係数が非ゼロである次数の最小値)
    int order() const {
        for (int d = 0; d < size(); d++) {
            unless(poly[d].is_zero()) { return d; }
        }
        return -1;
    }
};
#line 5 "/home/user/competitive_programming/library_for_cpp/Modulo_Polynomial/Numeric_Theory_Translation.hpp"

template<typename F>
class Numeric_Theory_Translation {
    public:
    F primitive;
    vector<F> root, iroot, rate2, irate2, rate3, irate3;

    public:
    Numeric_Theory_Translation() {
        primitive = primitive_root();
        build_up();
    }

    private:
    F primitive_root(){
        if (F::mod() == 2) { return F(1); }
        if (F::mod() == 998244353) { return F(3); }

        vector<int> fac;
        int v = F::mod() - 1;

        for (int q = 2; q * q <= v; q++){
            int e = 0;
            while (v % q == 0){
                e++; v /= q;
            }

            if (e > 0) { fac.emplace_back(q); }
        }

        if (v > 1) { fac.emplace_back(v); }

        F g(2);
        while (true) {
            bool flag = true;
            for (int q: fac) {
                if (pow(g, (F::mod() - 1) / q) == 1){
                    flag = false;
                    break;
                }
            }
            if (flag) { break; }
            g += 1;
        }
        return g;
    }

    void build_up() {
        int x = ~(F::mod() - 1) & (F::mod() - 2);
        int rank2 = bit_length(x);

        root.resize(rank2 + 1); iroot.resize(rank2 + 1);
        rate2.resize(max(0, rank2 - 1)); irate2.resize(max(0, rank2 - 1));
        rate3.resize(max(0, rank2 - 2)); irate3.resize(max(0, rank2 - 2));

        root.back() = pow(primitive, (F::mod() - 1) >> rank2);
        iroot.back() = root.back().inverse();

        for (int i = rank2 - 1; i >= 0; i--){
            root[i] = root[i + 1] * root[i + 1];
            iroot[i] = iroot[i + 1] * iroot[i + 1];
        }

        F prod(1), iprod(1);
        for (int i = 0; i < rank2 - 1; i++){
            rate2[i] = root[i + 2] * prod;
            irate2[i] = iroot[i + 2] * prod;

            prod *= iroot[i + 2]; iprod *= root[i + 2];
        }

        prod = 1; iprod = 1;
        for (int i = 0; i < rank2 - 2; i++){
            rate3[i] = root[i + 3] * prod;
            irate3[i] = iroot[i + 3] * iprod;

            prod *= iroot[i + 3]; iprod *= root[i + 3];
        }
    }

    public:
    void ntt(vector<F> &A){
        int N = A.size();
        int h = ceil_log2(N);

        F I = root[2];
        for (int l = 0; l < h;){
            if (h - l == 1){
                int p = 1 << (h - l - 1);
                F rot(1);
                for (int s = 0; s < (1 << l); s++){
                    int offset = s << (h - l);
                    for(int i = 0; i < p; i++){
                        F x = A[i + offset], y = A[i + offset + p] * rot;
                        A[i + offset] = x + y;
                        A[i + offset + p] = x - y;
                    }

                    unless (s + 1 == (1 << l)){ rot *= rate2[bit_length(~s & -(~s)) - 1]; }
                }
                l++;
            } else {
                int p = 1 << (h - l - 2);
                F rot(1);
                for (int s = 0; s < (1 << l); s++){
                    F rot2 = rot * rot, rot3 = rot2 * rot;
                    int offset = s << (h - l);
                    for (int i = 0; i < p; i++){
                        F a0 = A[i + offset];
                        F a1 = A[i + offset + p] * rot;
                        F a2 = A[i + offset + 2 * p] * rot2;
                        F a3 = A[i + offset + 3 * p] * rot3;

                        F alpha = (a1 - a3) * I;

                        A[i + offset]         = a0 + a2 + a1 + a3;
                        A[i + offset + p]     = a0 + a2 - a1 - a3;
                        A[i + offset + 2 * p] = a0 - a2 + alpha;
                        A[i + offset + 3 * p] = a0 - a2 - alpha;
                    }

                    unless(s + 1 == 1 << l) { rot *= rate3[bit_length(~s & -(~s)) - 1]; }
                }
                l += 2;
            }
        }
    }

    public:
    void inverse_ntt(vector<F> &A){
        int N = A.size();
        int h = ceil_log2(N);

        F J = iroot[2];
        for (int l = h; l > 0;){
            if (l == 1){
                int p = 1 << (h - l);
                F irot(1);
                for (int s = 0; s < (1 << (l - 1)); s++){
                    int offset = s << (h - l + 1);
                    for(int i = 0; i < p; i++){
                        F x = A[i + offset], y = A[i + offset + p];
                        A[i + offset]   = x + y;
                        A[i + offset + p] = (x - y) * irot;
                    }

                    unless (s+1 == 1 << (l - 1) ) { irot *= irate2[bit_length(~s & -(~s)) -1]; }
                }
                l--;
            } else {
                int p = 1 << (h - l);
                F irot(1);
                for (int s=0; s<(1<<(l-2)); s++){
                    F irot2 = irot * irot, irot3 = irot2  *irot;
                    int offset=s<<(h-l+2);
                    for (int i = 0; i < p; i++){
                        F a0 = A[i + offset];
                        F a1 = A[i + offset + p];
                        F a2 = A[i + offset + 2 * p];
                        F a3 = A[i + offset + 3 * p];

                        F beta = (a2 - a3) * J;

                        A[i + offset]         = a0 + a2 + a1 + a3;
                        A[i + offset + p]     = (a0 - a1 + beta) * irot;
                        A[i + offset + 2 * p] = (a0 + a1 - a2 - a3) * irot2;
                        A[i + offset + 3 * p] = (a0 - a1 - beta) * irot3;
                    }

                    unless (s + 1 == 1 << (l - 2)) { irot *= irate3[bit_length(~s & -(~s)) - 1]; }
                }
                l-=2;
            }
        }

        F N_inv=F(N).inverse();
        for (int i=0; i<N; i++) A[i]*=N_inv;
    }

    vector<F> convolution(vector<F> A, vector<F> B){
        if (A.empty() || B.empty()) return vector<F>{};

        int M=A.size(), N=B.size(), L=M+N-1;
        if (min(M,N)<64){
            vector<F> C(L);
            for(int i=0; i<M; i++){
                for (int j=0; j<N; j++){
                    C[i+j]+=A[i]*B[j];
                }
            }
            return C;
        }

        int h=bit_length(L);
        int K=1<<h;

        vector<F> X(K), Y(K);
        copy(A.begin(), A.end(), X.begin());
        copy(B.begin(), B.end(), Y.begin());

        ntt(X); ntt(Y);
        for (int i=0; i<K; i++) X[i]*=Y[i];

        inverse_ntt(X); X.resize(L);
        return X;
    }

    vector<F> inverse(vector<F> P, int d) {
        int n = P.size();
        assert(!P.empty() && !P[0].is_zero());

        vector<F> G{P[0].inverse()};
        while (G.size() < d) {
            int m = G.size();
            vector<F> A(P.begin(), P.begin() + min(n, 2 * m));
            A.resize(2 * m);
            vector<F> B(G);
            B.resize(2 * m);

            ntt(A); ntt(B);
            for (int i = 0; i < 2 * m; i++) { A[i] *= B[i]; }

            inverse_ntt(A);
            A.erase(A.begin(), A.begin() + m);
            A.resize(2 * m);

            ntt(A);
            for (int i = 0; i < 2 * m; i++) { A[i] *= -B[i]; }

            inverse_ntt(A);

            G.insert(G.end(), A.begin(), A.begin() + m);
        }

        G.resize(d);
        return G;
    }

    vector<F> inverse(vector<F> P) { return inverse(P, P.size()); }

    vector<F> multiple_convolution(vector<vector<F>> A) {
        if (A.empty()) { return {1}; }

        deque<int> queue(A.size());
        iota(queue.begin(), queue.end(), 0);

        while (queue.size() > 1) {
            int i = queue.front(); queue.pop_front();
            int j = queue.front(); queue.pop_front();

            A[i] = convolution(A[i], A[j]);
            queue.emplace_back(i);
        }

        return A[queue.back()];
    }
};
#line 2 "/home/user/competitive_programming/library_for_cpp/Binary_Search/General_Integer.hpp"

// [L, R] 上で広義単調増加な条件 cond に対して, cond(x) が True になる最小の整数 x を二分探索で求める.
// Args
// T L: 下限
// T R: 上限
// function<bool(T)> cond: [L, R] 上広義単調増加な条件
// T default_value: cond(R) が False の時の返り値
template<typename T>
T General_Binary_Increase_Search_Integer(T L, T R, const function<bool(T)> cond, T default_value) {
    // 例外ケースの処理
    // R でも False → 異常値
    unless(cond(R)) { return default_value; }
    // L にて True → L
    if(cond(L)) { return L; }

    // 探索パート
    while (R - L > 1) {
        T C = L + (R - L) / 2;
        cond(C) ? R = C : L = C;
    }

    return R;
}

// [L, R] 上で広義単調減少な条件 cond に対して, cond(x) が True になる最大の整数 x を二分探索で求める.
// Args
// T L: 下限
// T R: 上限
// function<bool(T)> cond: [L, R] 上広義単調減少な条件
// T default_value: cond(L) が False の時の返り値
template<typename T>
T General_Binary_Decrease_Search_Integer(T L, T R, const function<bool(T)> cond, T default_value) {
    // 例外ケースの処理
    // L でも False → 異常値
    unless(cond(L)) { return default_value; }
    // R にて True → R
    if(cond(R)) { return R; }

    // 探索パート
    while (R - L > 1) {
        T C = L + (R - L) / 2;
        cond(C) ? L = C : R = C;
    }

    return L;
}
#line 8 "program.cpp"

const ll Mod = 998244353;
using mint = modint<998244353>;
auto calculator = Numeric_Theory_Translation<mint>();

template<typename T, typename U, typename FUNC>
vector<U> vector_mapping(const vector<T> &vec, const FUNC &func) {
    int n = vec.size();
    vector<U> res(n);
    for (int i = 0; i < n; i++) {
        res[i] = func(vec[i]);
    }

    return res;
}

using M = pair<int, int>;

vector<vector<mint>> solve() {
    int N, Q; cin >> N >> Q;
    string S; cin >> S; S = "*" + S;
    auto W = vector_input<ll>(N + 1, 0);

    S[N] = 'B';

    auto X = Lazy_Segment_Tree<M, int> (
        vector_mapping<char, M>(
            vector(S.begin(), S.end()),
            [](const char c) -> M { return c == 'G' ? make_pair(1, 0) : make_pair(0, 1); }
        ),
        [](const M x, const M y) -> M { return { x.first + y.first, x.second + y.second }; },
        {0, 0},
        [](const int a, const M x) -> M { return { safe_mod(a, 2) == 0 ? x : make_pair(x.second, x.first) }; },
        add<int>,
        0
    );

    auto W_sum = Range_Add_Range_Sum_Lazy_Segment_Tree<ll>(W);

    auto latest_backed_at = [&X, &N](const int v) -> int {
        M alpha = X.product(1, v - 1);
        if (alpha.second == 0) { return 0; }

        return X.max_right(1, [&alpha](const M z) -> bool { return z.second < alpha.second; });
    };

    auto next_back_at = [&X, &N](const int v) -> int {
        M beta = X.product(1, v - 1);
        return min(X.max_right(1, [&beta](const M z) -> bool { return z.second <= beta.second; }), N);
    };

    vector<vector<mint>> ans;
    for (int q = 1; q <= Q; ++q) {
        int t; cin >> t;
        if (t == 1) {
            int l, r; cin >> l >> r;
            if (r == N) r--;
            X.action(l, r, 1);
        } else if (t == 2) {
            int l, r, a; cin >> l >> r >> a;
            W_sum.action(l, r, a);
        } else if (t == 3) {
            int v, K; cin >> v >> K;
            auto U = vector_input<int>(K, 0);

            int r = latest_backed_at(v);
            unordered_map<int, int> back_time;
            int less = 0;
            for (const int u: U) {
                if (u < v) { less++; continue; }

                int t = latest_backed_at(u);
                back_time[t]++;
            }

            int vr = next_back_at(v);
            mint p = mint(W_sum.sum(v, vr)) / mint(W_sum.sum(0, vr));

            vector<vector<mint>> polys;
            vector<mint> poly(back_time[r] + 1 + less, 0);
            poly[back_time[r]] = 1;

            polys.emplace_back(poly);
            for (auto &&[t, d]: back_time) {
                if (t == r) continue;

                vector<mint> poly(d + 1, 0);
                poly[0] += 1 - p;
                poly[d] += p;

                polys.emplace_back(poly);
            }

            auto P = calculator.multiple_convolution(polys);
            ans.emplace_back(P);
        }
    }

    return ans;
}

int main() {
    for (auto ans: solve()) {
        cout << ans << "\n";
    }
}