/**
 * date   : 2026-10-03 00:21:55
 * author : yamadanull
 */


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

namespace yamada {
using ll = long long;
using i32 = int;
using u32 = unsigned int;
using i64 = long long;
using u64 = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
using lld = long double;

template <typename T>
using vc = vector<T>;
template <typename T>
using VV = vector<vector<T>>;
template <typename T>
using VVV = vector<vector<vector<T>>>;
template <typename T>
using VVVV = vector<vector<vector<vector<T>>>>;
using vl = vector<long long>;
using vd = vector<double>;
using vs = vector<string>;
using vb = vector<bool>;
using vvl = vector<vector<long long>>;
using vvvl = vector<vector<vector<long long>>>;
using vvvvl = vector<vector<vector<vector<long long>>>>;
template <typename T>
using minpq = priority_queue<T, vector<T>, greater<T>>;
template <typename T>
using maxpq = priority_queue<T, vector<T>, less<T>>;

template <typename T, typename U>
struct pr : pair<T, U> {
	template <typename... Args>
	pr(Args... args) : pair<T, U>(args...) {}

	using pair<T, U>::first;
	using pair<T, U>::second;

	pr &operator+=(const pr &r) {
		first += r.first;
		second += r.second;
		return *this;
	}
	pr &operator-=(const pr &r) {
		first -= r.first;
		second -= r.second;
		return *this;
	}
	pr &operator*=(const pr &r) {
		first *= r.first;
		second *= r.second;
		return *this;
	}
	template <typename S>
	pr &operator*=(const S &r) {
		first *= r, second *= r;
		return *this;
	}
	pr operator+(const pr &r) const { return pr(*this) += r; }
	pr operator-(const pr &r) const { return pr(*this) -= r; }
	pr operator*(const pr &r) const { return pr(*this) *= r; }
	template <typename S>
	pr operator*(const S &r) const {
		return pr(*this) *= r;
	}
	pr operator-() const { return pr{-first, -second}; }
};

using pl = pr<ll, ll>;
using vp = vc<pl>;
using vvp = VV<pl>;

constexpr int inf = 1001001001;
constexpr long long infLL = 4004004004004004004LL;

template <typename T>
int len(const T &t) { return t.size(); }

template <typename T, typename U>
inline bool amin(T &x, U y) { return (y < x) ? (x = y, true) : false; }
template <typename T, typename U>
inline bool amax(T &x, U y) { return (x < y) ? (x = y, true) : false; }

template <typename T>
inline long long Sum(const vector<T> &v) { return accumulate(v.begin(), v.end(), T(0)); }

template <typename T>
int lb(const vector<T> &v, const T &a) { return ranges::lower_bound(v, a) - v.begin(); }
template <typename T>
int ub(const vector<T> &v, const T &a) { return ranges::upper_bound(v, a) - v.begin(); }

constexpr long long TEN(int n) {
	long long ret = 1, x = 10;
	for (; n; x *= x, n >>= 1) ret *= (n & 1 ? x : 1);
	return ret;
}

template <typename T>
vector<T> mkrui(const vector<T> &v, bool rev = false) {
	vector<T> ret(v.size() + 1);
	if (rev) {
		for (int i = int(v.size()) - 1; i >= 0; i--) ret[i] = v[i] + ret[i + 1];
	} else {
		for (int i = 0; i < int(v.size()); i++) ret[i + 1] = ret[i] + v[i];
	}
	return ret;
};

template <typename T>
vector<T> mkuni(const vector<T> &v) {
	vector<T> ret(v);
	ranges::sort(ret);
	ret.erase(ranges::unique(ret).begin(), ret.end());
	return ret;
}

template <typename F>
vector<int> mkord(int N, F f) {
	vector<int> ord = views::iota(0, N) | ranges::to<vector>();
	ranges::sort(ord, f);
	return ord;
}

template <typename T>
vector<int> mkinv(vector<T> &v) {
	vector<int> inv(ranges::max(v) + 1, -1);
	for (int i = 0; i < (int)v.size(); i++) inv[v[i]] = i;
	return inv;
}

vector<int> mkiota(int N) { return views::iota(0, N) | ranges::to<vector>(); }

// g o f
vector<int> p_mrg(vector<int> f, const vector<int> &g) {
	for (int i = 0; i < (int)f.size(); i++) f[i] = g[f[i]];
	return f;
}

// f: old_idx -> new_idx
template <typename T>
vector<T> p_shf(const vector<int>& f, const vector<T>& A, const bool inv = false) {
	int n = A.size();
	vector<T> ret(n);
	if (!inv) for (int i = 0; i < n; i++) ret[f[i]] = A[i];
	else for (int i = 0; i < n; i++) ret[i] = A[f[i]];
	return ret;
}

template <typename T>
T mkrev(const T &v) { return v | views::reverse | ranges::to<vector>(); }

template <typename C>
bool nxp(C &v) { return ranges::next_permutation(v).found; }
template <typename F>
void nxp(int N, F f) {
	vector<int> a = views::iota(0, N) | ranges::to<vector>();
	do { f(a); } while (nxp(a));
}

// 返り値の型は入力の T に依存
// i 要素目 : [0, a[i])
template <typename T>
vector<vector<T>> product(const vector<T> &a) {
	vector<vector<T>> ret;
	vector<T> v;
	auto dfs = [&](auto rc, int i) -> void {
		if (i == (int)a.size()) { ret.emplace_back(v); return; }
		for (int j = 0; j < a[i]; j++) v.emplace_back(j), rc(rc, i + 1), v.pop_back();
	}; dfs(dfs, 0);
	return ret;
}

template <typename T, typename U>
vector<U> Digit(T a, const U &x, int siz = -1) {
	vector<U> ret;
	while (a > 0) {
		ret.emplace_back(a % x);
		a /= x;
	}
	if (siz >= 0) ret.resize(siz);
	return ret;
}

// F : function(void(T&)), mod を取る操作
// T : 整数型のときはオーバーフローに注意する
template <typename T>
T Power(T a, long long n, const T &I, const function<void(T &)> &f) {
	T res = I;
	for (; n; f(a = a * a), n >>= 1) {
		if (n & 1) f(res = res * a);
	}
	return res;
}
// T : 整数型のときはオーバーフローに注意する
template <typename T>
T Power(T a, long long n, const T &I = T{1}) {
	return Power(a, n, I, function<void(T &)>{[](T &) -> void {}});
}

template <typename T>
vector<T> Transpose(const vector<T> &v) {
	using U = typename T::value_type;
	if(v.empty()) return {};
	int H = v.size(), W = v[0].size();
	vector res(W, T(H, U{}));
	for (int i = 0; i < H; i++) for (int j = 0; j < W; j++) res[j][i] = v[i][j];
	return res;
}

template <typename T>
vector<T> Rotate(const vector<T> &v, int clockwise = true) {
	using U = typename T::value_type;
	int H = v.size(), W = v[0].size();
	vector res(W, T(H, U{}));
	for (int i = 0; i < H; i++) for (int j = 0; j < W; j++) {
		if (clockwise) res[W - 1 - j][i] = v[i][j];
		else res[j][H - 1 - i] = v[i][j];
	}
	return res;
}

template <typename T, typename F>
T bisect(T ok, T bad, F pred) {
	if (ok == bad) return ok;
	while (bad - ok > 1) { T mid = ok + (bad - ok) / 2; (pred(mid) ? ok : bad) = mid; } 
	return bad;
}

template <typename T, typename F>
T bisect_double(T ok, T bad, F pred, int iter = 100) {
	if (ok == bad) return ok;
	while (iter--) { T mid = ok + (bad - ok) / 2; (pred(mid) ? ok : bad) = mid; } 
	return bad;
}

template <typename T0, typename T1, typename T2>
bool inLR(T0 L, T1 x, T2 R){ return L <= x && x < R; }
bool YESNO(bool b) { cout << (b ? "YES\n" : "NO\n"); return b; }
bool YesNo(bool b) { cout << (b ? "Yes\n" : "No\n"); return b; }
bool yesno(bool b) { cout << (b ? "yes\n" : "no\n"); return b; }

bool is_square(uint64_t n) {
	if (n < 2) return true;
	uint64_t r = static_cast<uint64_t>(sqrtl(static_cast<long double>(n)));
	if (r * r == n) return true;
	++r;
	return r * r == n;
}

long long Isqrt(long long n) {
	if (n <= 0) return 0;
	long long x = sqrt(n);
	while ((x + 1) * (x + 1) <= n) x++;
	while (x * x > n) x--;
	return x;
}

template <typename T>
struct CumulativeSum {
	vector<T> S;
	CumulativeSum(vector<T> &A) {
		int N = A.size();
		S.resize(N + 1);
		for (int i = 0; i < N; i++) S[i + 1] = S[i] + A[i];
	}
	T query(int l, int r) { return (l <= r ? S[r] - S[l] : (T)0); }
	T query() { return S.back(); }
	inline T operator()(int l, int r) { return query(l, r); }
	inline T operator()() { return query(); }
};

long long Floor(long long a, long long b) {
	assert(b != 0);
	if (b < 0) a = -a, b = -b;
	return a / b - (a % b < 0);
}
long long Under(long long a, long long b) {
	assert(b != 0);
	if (b < 0) a = -a, b = -b;
	return a / b - (a % b <= 0);
}
long long Ceil(long long a, long long b) {
	assert(b != 0);
	if (b < 0) a = -a, b = -b;
	return a / b + (a % b > 0);
}
long long Over(long long a, long long b) {
	assert(b != 0);
	if (b < 0) a = -a, b = -b;
	return a / b + (a % b >= 0);
}
long long Modulo(long long a, long long b) {
	assert(b > 0);
	long long c = a % b;
	return c < 0 ? c + b : c;
}

using ranges::views::enumerate;
using ranges::views::take;
using ranges::views::take_while;
using ranges::views::drop;
using ranges::views::drop_while;
using ranges::views::filter;
using ranges::views::join;
using ranges::views::adjacent;
using ranges::views::slide;
using ranges::views::stride;
using ranges::views::chunk;
using ranges::views::zip;

constexpr auto Sort = ranges::sort;
constexpr auto Rev = ranges::reverse;
constexpr auto Min = ranges::min;
constexpr auto Max = ranges::max;
constexpr auto Bisect = ranges::binary_search;
constexpr auto rtov = ranges::to<vector>();
constexpr auto rtos = ranges::to<string>();

constexpr auto vwrev = ranges::views::reverse;
constexpr auto vwtrans = ranges::views::transform;
constexpr auto vwiota = ranges::views::iota;

namespace rg = ranges;
namespace vw = views;
using namespace placeholders;
} // namespace yamada

namespace yamada {
__attribute__((target("popcnt"))) inline int popcnt(const u64 &a) {
	return __builtin_popcountll(a);
}
inline int lsb(const u64 &a) { return a ? __builtin_ctzll(a) : 64; }
inline int msb(const u64 &a) { return a ? 63 - __builtin_clzll(a) : -1; }
template <typename T>
inline int gbit(const T &a, int i) { return (a >> i) & 1; }
template <typename T>
inline void sbit(T &a, int i, bool b) { if (gbit(a, i) != b) a ^= T(1) << i; }
constexpr long long PW(int n) { return 1LL << n; }
constexpr long long MSK(int n) { return (1LL << n) - 1; }
}  // namespace yamada
namespace yamada {

template <typename T, size_t K> ostream &operator<<(ostream &os, const array<T, K> &v);
template <typename T, size_t K> istream &operator>>(istream &is, array<T, K> &v);
template <typename T> ostream &operator<<(ostream &os, const vector<T> &v);
template <typename T> istream &operator>>(istream &is, vector<T> &v);
template <typename T, typename U> ostream &operator<<(ostream &os, const pair<T, U> &p);
template <typename T, typename U> istream &operator>>(istream &is, pair<T, U> &p);

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

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

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

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

template <typename T, size_t K>
ostream &operator<<(ostream &os, const array<T, K> &v) {
	for (int i = 0; i < K; i++) os << (i ? " " : "") << v[i];
	return os;
}

template <typename T, size_t K>
istream &operator>>(istream &is, array<T, K> &v) {
	for (auto &x : v) is >> x;
	return is;
}

istream &operator>>(istream &is, __int128_t &x) {
	string S;
	is >> S;
	x = 0;
	int flag = 0;
	for (auto &c : S) {
		if (c == '-') {
			flag = true;
			continue;
		}
		x *= 10;
		x += c - '0';
	}
	if (flag) x = -x;
	return is;
}

istream &operator>>(istream &is, __uint128_t &x) {
	string S;
	is >> S;
	x = 0;
	for (auto &c : S) {
		x *= 10;
		x += c - '0';
	}
	return is;
}

ostream &operator<<(ostream &os, __int128_t x) {
	if (x == 0) return os << 0;
	if (x < 0) os << '-', x = -x;
	string S;
	while (x) S.push_back('0' + x % 10), x /= 10;
	reverse(begin(S), end(S));
	return os << S;
}
ostream &operator<<(ostream &os, __uint128_t x) {
	if (x == 0) return os << 0;
	string S;
	while (x) S.push_back('0' + x % 10), x /= 10;
	reverse(begin(S), end(S));
	return os << S;
}

void in() {}
template <typename T, class... U>
void in(T &t, U &...u) {
	cin >> t;
	in(u...);
}

void out() { cout << "\n"; }
template <char sep = ' ', typename T, class... U>
void out(const T &t, const U &...u) {
	cout << t;
	if constexpr (sizeof...(u)) {
		cout << sep;
		out<sep>(u...);
	}
	else out();
}

void fout() { cout << endl; }
template <char sep = ' ', typename T, class... U>
void fout(const T &t, const U &...u) {
	cout << t;
	if constexpr (sizeof...(u)) {
		cout << sep;
		fout<sep>(u...);
	}
	else fout();
}

void wout() {}
template <char sep = ' ', typename T, class... U>
void wout(const T &t, const U &...u) {
	cout << t;
	if constexpr (sizeof...(u)) {
		cout << sep;
		wout<sep>(u...);
	}
	else wout();
}

template <size_t iter = 1000, typename mint>
std::string toFraction(const mint &a) {
	for (int deno = 1; deno <= iter; deno++) {
		mint inv = ((mint)deno).inverse();
		for (int nume = 0; nume <= iter; nume++) {
			mint val = inv * nume;
			if (val == a) {
				if (deno == 1) return std::to_string(nume);
				return std::to_string(nume) + "/" + std::to_string(deno);
			}
			else if (-val == a) {
				if (deno == 1) return std::to_string(-nume);
				return std::to_string(-nume) + "/" + std::to_string(deno);
			}
		}
	}
	return "NF";
}

void mout() { cout << endl; }
template <size_t iter = 1000, char sep = ' ', typename mint, class... U>
void mout(const mint &a, const U &...u) {
	std::cout << toFraction<iter, mint>(a);
	if constexpr (sizeof...(u)) {
		cout << sep;
		mout<iter, sep>(u...);
	}
	else mout();
}

template <size_t iter = 1000, typename mint>
void mout(std::vector<mint> &A) {
	for (int i = 0; i < (int)A.size(); i++) {
		std::cout << toFraction(A[i], iter) << (i == (int)A.size() - 1 ? "\n" : " ");
	}
}

struct IoSetupYamada {
	IoSetupYamada() {
		cin.tie(nullptr);
		ios::sync_with_stdio(false);
		cout << fixed << setprecision(15);
		cerr << fixed << setprecision(7);
	}
} iosetupyamada;

}  // namespace yamada
namespace DebugImpl {

template <typename U, typename = void>
struct is_specialize : false_type {};
template <typename U>
struct is_specialize< U, typename conditional<false, typename U::iterator, void>::type> : true_type {};
template <typename U>
struct is_specialize< U, typename conditional<false, decltype(U::first), void>::type> : true_type {};
template <typename U>
struct is_specialize<U, enable_if_t<is_integral<U>::value, void>> : true_type {};

void dump(const char& t) { cerr << t; }

void dump(const string& t) { cerr << t; }

void dump(const bool& t) { cerr << (t ? "true" : "false"); }

void dump(__int128_t t) {
	if (t == 0) cerr << 0;
	if (t < 0) cerr << '-', t = -t;
	string S;
	while (t) S.push_back('0' + t % 10), t /= 10;
	reverse(begin(S), end(S));
	cerr << S;
}

void dump(__uint128_t t) {
	if (t == 0) cerr << 0;
	string S;
	while (t) S.push_back('0' + t % 10), t /= 10;
	reverse(begin(S), end(S));
	cerr << S;
}

template <typename U, enable_if_t<!is_specialize<U>::value, nullptr_t> = nullptr>
void dump(const U& t) { cerr << t; }

template <typename T>
void dump(const T& t, enable_if_t<is_integral<T>::value>* = nullptr) {
	string res;
	if (t == yamada::inf) res = "inf";
	if constexpr (is_signed<T>::value) {
		if (t == -yamada::inf) res = "-inf";
	}
	if constexpr (sizeof(T) == 8) {
		if (t == yamada::infLL) res = "inf";
		if constexpr (is_signed<T>::value) {
			if (t == -yamada::infLL) res = "-inf";
		}
	}
	if (res.empty()) res = to_string(t);
	cerr << res;
}

template <typename T, typename U>
void dump(const pair<T, U>&);
template <typename T>
void dump(const pair<T*, int>&);

template <typename T>
void dump(const T& t, enable_if_t<!is_void<typename T::iterator>::value>* = nullptr) {
	cerr << "[ ";
	for (auto it = t.begin(); it != t.end();) {
		dump(*it);
		cerr << (++it == t.end() ? "" : ", ");
	}
	cerr << " ]";
}

template <typename T, typename U>
void dump(const pair<T, U>& t) {
	cerr << "( ";
	dump(t.first);
	cerr << ", ";
	dump(t.second);
	cerr << " )";
}

template <typename T>
void dump(const pair<T*, int>& t) {
	cerr << "[ ";
	for (int i = 0; i < t.second; i++) {
		dump(t.first[i]);
		cerr << (i == t.second - 1 ? "" : ", ");
	}
	cerr << " ]";
}

void trace() { cerr << endl; }
template <typename Head, typename... Tail>
void trace(Head&& head, Tail&&... tail) {
	cerr << " ";
	dump(head);
	if (sizeof...(tail) != 0) cerr << ",";
	trace(std::forward<Tail>(tail)...);
}

}  // namespace DebugImpl


#ifdef yamadaDebug
#define err(...) \
	do { \
		cerr << "## " << #__VA_ARGS__ << " = "; \
		DebugImpl::trace(__VA_ARGS__); \
	} while (0)
#else
#define err(...) (void(0))
#endif

#ifdef yamadaLocal
#define err2(...) \
	do { \
		cerr << "## " << #__VA_ARGS__ << " = "; \
		DebugImpl::trace(__VA_ARGS__); \
	} while (0)
#else
#define err2(...) (void(0))
#endif

#define each1(x, v) for (auto&& x : v)
#define each2(x, y, v) for (auto&& [x, y] : v)
#define each3(x, y, z, v) for (auto&& [x, y, z] : v)
#define each4(x, y, z, w, v) for (auto&& [x, y, z, w] : v)
#define overload5(a, b, c, d, e, f, ...) f
#define each(...) overload5(__VA_ARGS__, each4, each3, each2, each1)(__VA_ARGS__)

#define rep1(a) for (long long _ = 0; _ < (long long)(a); ++_)
#define rep2(i, a) for (long long i = 0; i < (long long)(a); ++i)
#define rep3(i, a, b) for (long long i = a; i < (long long)(b); ++i)
#define rep4(i, a, b, c) for (long long i = a; i < (long long)(b); i += c)
#define overload4(a, b, c, d, e, ...) e
#define rep(...) overload4(__VA_ARGS__, rep4, rep3, rep2, rep1)(__VA_ARGS__)

#define rep1r(a) for (long long i = (long long)(a)-1; i >= 0LL; --i)
#define rep2r(i, a) for (long long i = (long long)(a)-1; i >= 0LL; --i)
#define rep3r(i, a, b) for (long long i = (long long)(b)-1; i >= (long long)(a); --i)
#define overload3(a, b, c, d, ...) d
#define repr(...) overload3(__VA_ARGS__, rep3r, rep2r, rep1r)(__VA_ARGS__)

#define eb emplace_back
#define pb push_back
#define mkp make_pair
#define mkt make_tuple
#define fi first
#define se second

#define vv(type, name, h, ...)  \
	vector<vector<type> > name(h, vector<type>(__VA_ARGS__))
#define vvv(type, name, h, w, ...) \
	vector<vector<vector<type>>> name( \
			h, vector<vector<type>>(w, vector<type>(__VA_ARGS__)))
#define vvvv(type, name, a, b, c, ...)  \
	vector<vector<vector<vector<type>>>> name( \
			a, vector<vector<vector<type>>>( \
				b, vector<vector<type>>(c, vector<type>(__VA_ARGS__))))

#define ini(...) \
	int __VA_ARGS__; \
	in(__VA_ARGS__)
#define inl(...) \
	long long __VA_ARGS__; \
	in(__VA_ARGS__)
#define ins(...) \
	string __VA_ARGS__; \
	in(__VA_ARGS__)
#define in2(s, t) \
	for (int i = 0; i < (int)s.size(); i++) { \
		in(s[i], t[i]); \
	}
#define in3(s, t, u) \
	for (int i = 0; i < (int)s.size(); i++) { \
		in(s[i], t[i], u[i]); \
	}
#define in4(s, t, u, v) \
	for (int i = 0; i < (int)s.size(); i++) { \
		in(s[i], t[i], u[i], v[i]); \
	}
#define die(...) \
	do { \
		yamada::out(__VA_ARGS__);\
		return; \
	} while (0)

namespace yamada {
void solve();
}
int main() { yamada::solve(); }



namespace MONOID {

template <typename E>
struct Add {
	using value_type = E;
	static constexpr value_type O(const value_type& x, const value_type& y) noexcept { return x + y; }
	static constexpr value_type inverse(const value_type &x) noexcept { return -x; }
	static constexpr value_type power(const value_type& x, long long n) noexcept { return value_type(n) * x; }
	static constexpr value_type I() { return value_type(0); }
	static constexpr bool COMMUTATIVE = true;
};

} // namespace monoid

#include <algorithm>
#include <limits>

namespace MONOID {

template <typename E, E MNF = std::numeric_limits<E>::min()>
struct Max {
	using value_type = E;
	static constexpr value_type O(const value_type& x, const value_type& y) { return std::max(x, y); }
	static constexpr value_type I() { return MNF; }
	static constexpr bool COMMUTATIVE = true;
};

} // namespace monoid

namespace ACTED_MONOID {

template <typename E, E MNF = std::numeric_limits<E>::min()>
struct Add_Max {
	using Monoid_X = MONOID::Max<E, MNF>;
	using Monoid_A = MONOID::Add<E>;
	using X = typename Monoid_X::value_type; // E
	using A = typename Monoid_A::value_type; // E
	static constexpr X O(const X& x, const A& a, const long long) {
		return x == MNF ? x : x + a;
	}
};

} // namespace ACTED_MONOID



#include <algorithm>
#include <limits>

namespace MONOID {

template <typename E, E INF = std::numeric_limits<E>::max()>
struct Min {
	using value_type = E;
	static constexpr value_type O(const value_type& x, const value_type& y) { return std::min(x, y); }
	static constexpr value_type I() { return INF; }
	static constexpr bool COMMUTATIVE = true;
};

} // namespace monoid

namespace ACTED_MONOID {

template <typename E, E INF = std::numeric_limits<E>::max()>
struct Add_Min {
	using Monoid_X = MONOID::Min<E, INF>;
	using Monoid_A = MONOID::Add<E>;
	using X = typename Monoid_X::value_type; // E
	using A = typename Monoid_A::value_type; // E
	static constexpr X O(const X& x, const A& a, const long long) { return x + a; }
};

} // namespace ACTED_MONOID



namespace ACTED_MONOID {

template <typename E>
struct Add_Sum {
	using Monoid_X = MONOID::Add<E>;
	using Monoid_A = MONOID::Add<E>;
	using X = typename Monoid_X::value_type; // E
	using A = typename Monoid_A::value_type; // E
	static constexpr X O(const X& x, const A& a, const long long &size) {
		return x + a * E(size);
	}
};

} // namespace ACTED_MONOID



#include <optional>

namespace MONOID {

template <typename T>
struct Update {
	using value_type = std::optional<T>;
	static constexpr value_type O(const value_type& x, const value_type& y) { return y.has_value() ? y : x; }
	static constexpr value_type I() { return std::nullopt; }
	static constexpr bool COMMUTATIVE = false;
};

} // namespace monoid

namespace ACTED_MONOID {

template <typename E, E MNF = numeric_limits<E>::min()>
struct Update_Max {
	using Monoid_X = MONOID::Max<E, MNF>;
	using Monoid_A = MONOID::Update<E>;
	using X = typename Monoid_X::value_type; // E
	using A = typename Monoid_A::value_type; // optional<E>
	static constexpr X O(const X& x, A& a, const long long) {
		return a.has_value() ? a.value() : x;
	}
};

} // namespace ACTED_MONOID



namespace ACTED_MONOID {

template <typename E, E INF = std::numeric_limits<E>::max()>
struct Update_Min {
	using Monoid_X = MONOID::Min<E, INF>;
	using Monoid_A = MONOID::Update<E>;
	using X = typename Monoid_X::value_type; // E
	using A = typename Monoid_A::value_type; // optional<E>
	static constexpr X O(const X& x, const A& a, const long long) {
		return a.has_value() ? a : x;
	}
};

} // namespace ACTED_MONOID



namespace ACTED_MONOID {

template <typename E>
struct Update_Sum {
	using Monoid_X = MONOID::Add<E>;
	using Monoid_A = MONOID::Update<E>;
	using X = typename Monoid_X::value_type; // E
	using A = typename Monoid_A::value_type; // optional<E>
	static constexpr X O(const X& x, const A& a, const long long &size) {
		return a.has_value() ? a.value() * E(size) : x;
	}
};

} // namespace ACTED_MONOID


template <typename Monoid>
struct SegmentTree {
	using X = typename Monoid::value_type;
	std::vector<X> dat;
	int N, LOG, size;

	SegmentTree() {}
	SegmentTree(int N) { build(N); }
	template <typename F>
	SegmentTree(int N, F init) { build(N, init); }
	template <typename T>
	SegmentTree(const std::vector<T>& v) { build(v); }

	void build(int m) {
		build(m, [](int i) -> X { return Monoid::I(); });
	}
	template <typename T>
	void build(const std::vector<T>& v) {
		build(v.size(), [&](int i) -> X { return v[i]; });
	}
	template <typename F>
	void build(int m, F init) {
		N = m, LOG = 1;
		while ((1 << LOG) < N) ++LOG;
		size = 1 << LOG;
		/* yamada::out("size:",size); */
		dat.assign(size << 1, Monoid::I());
		for (int i = 0; i < N; i++) dat[size + i] = init(i);
		for (int i = size - 1; i >= 1; i--) {
			/* yamada::out("i:",i); */
			update(i);
		}
	}

	X get(int i) const { return dat[size + i]; }
	std::vector<X> get() const { return {dat.begin() + size, dat.begin() + size + N}; }

	void update(int i) { dat[i] = Monoid::O(dat[2 * i], dat[2 * i + 1]); }
	void set(int i, const X x) {
		assert(i < N);
		dat[i += size] = x;
		while (i >>= 1) update(i);
	}

	void multiply(int i, const X x) {
		assert(i < N);
		i += size;
		dat[i] = Monoid::O(dat[i], x);
		while (i >>= 1) update(i);
	}

	X prod(int L, int R) const {
		assert(0 <= L && L <= R && R <= N);
		X vl = Monoid::I(), vr = Monoid::I();
		L += size, R += size;
		while (L < R) {
			if (L & 1) vl = Monoid::O(vl, dat[L++]);
			if (R & 1) vr = Monoid::O(dat[--R], vr);
			L >>= 1, R >>= 1;
		}
		return Monoid::O(vl, vr);
	}

	std::vector<int> prod_ids(int L, int R) const {
		assert(0 <= L && L <= R && R <= N);
		std::vector<int> I, J;
		L += size, R += size;
		while (L < R) {
			if (L & 1) I.eb(L++);
			if (R & 1) J.eb(--R);
			L >>= 1, R >>= 1;
		}
		std::reverse(J.begin(), J.end());
		I.resize(I.size() + J.size());
		for (int j = 0; j < (int)J.size(); j++) I[(int)I.size() - (int)J.size() + j] = J[j];
		return I;
	}

	X prod() const { return dat[1]; }
	template <typename... Args>
	X operator()(Args... args) { return prod(args...); }
	template <typename... Args>
	X operator[](Args... args) { return get(args...); }

	template <typename F>
	int max_right(int L, F check) const {
		assert(0 <= L && L <= N && check(Monoid::I()));
		if (L == N) return N;
		L += size;
		X sm = Monoid::I();
		do {
			while (L % 2 == 0) L >>= 1;
			if (!check(Monoid::O(sm, dat[L]))) {
				while (L < size) {
					L = 2 * L;
					if (check(Monoid::O(sm, dat[L]))) {
						sm = Monoid::O(sm, dat[L++]);
					}
				}
				return L - size;
			}
			sm = Monoid::O(sm, dat[L++]);
		} while ((L & -L) != L);
		return N;
	}

	template <typename F>
	int min_left(int R, F check) const {
		assert(0 <= R && R <= N && check(Monoid::I()));
		if (R == 0) return 0;
		R += size;
		X sm = Monoid::I();
		do {
			--R;
			while (R > 1 && (R % 2)) R >>= 1;
			if (!check(Monoid::O(dat[R], sm))) {
				while (R < size) {
					R = 2 * R + 1;
					if (check(Monoid::O(dat[R], sm))) {
						sm = Monoid::O(dat[R--], sm);
					}
				}
				return R + 1 - size;
			}
			sm = Monoid::O(dat[R], sm);
		} while ((R & -R) != R);
		return 0;
	}

	// prod_{l<=i<r} A[i xor x]
	X xor_prod(int l, int r, int xor_val) const {
		static_assert(Monoid::COMMUTATIVE);
		X x = Monoid::I();
		for (int k = 0; k < LOG + 1; ++k) {
			if (l >= r) break;
			if (l & 1) {
				x = Monoid::O(x, dat[(size >> k) + ((l++) ^ xor_val)]);
			}
			if (r & 1) {
				x = Monoid::O(x, dat[(size >> k) + ((--r) ^ xor_val)]);
			}
			l /= 2, r /= 2, xor_val /= 2;
		}
		return x;
	}
};

template <typename ActedMonoid>
struct LazySegmentTree {
	using MX = typename ActedMonoid::Monoid_X;
	using MA = typename ActedMonoid::Monoid_A;
	using X = typename MX::value_type;
	using A = typename MA::value_type;
	int N, LOG, size;
	std::vector<X> dat;
	std::vector<A> laz;
	std::vector<bool> has_laz;

	LazySegmentTree() {}
	LazySegmentTree(int N) { build(N); }
	template <typename F>
	LazySegmentTree(int N, F init) { build(N, init); }
	template <typename T>
	LazySegmentTree(const std::vector<T>& v) { build(v); }

	void build(int m) {
		build(m, [](int i) -> X { return MX::I(); });
	}
	template <typename T>
	void build(const std::vector<T>& v) {
		build(v.size(), [&](int i) -> X { return v[i]; });
	}
	template <typename F>
	void build(int m, F init) {
		N = m, LOG = 1;
		while ((1 << LOG) < N) ++LOG;
		size = 1 << LOG;
		dat.assign(size << 1, MX::I());
		laz.assign(size, MA::I());
		has_laz.assign(size, false);
		for (int i = 0; i < N; i++) dat[size + i] = init(i);
		for (int i = size - 1; i >= 1; i--) update(i);
	}

	void update(int k) { dat[k] = MX::O(dat[2 * k], dat[2 * k + 1]); }
	void set(int p, X x) {
		assert(0 <= p && p < N);
		p += size;
		for (int i = LOG; i >= 1; i--) push(p >> i);
		dat[p] = x;
		for (int i = 1; i <= LOG; i++) update(p >> i);
	}
	void multiply(int p, const X& x) {
		assert(0 <= p && p < N);
		p += size;
		for (int i = LOG; i >= 1; i--) push(p >> i);
		dat[p] = MX::O(dat[p], x);
		for (int i = 1; i <= LOG; i++) update(p >> i);
	}

	X get(int p) {
		assert(0 <= p && p < N);
		p += size;
		for (int i = LOG; i >= 1; i--) push(p >> i);
		return dat[p];
	}

	std::vector<X> get() {
		for (int k = 0; k < size; k++) push(k);
		return {dat.begin() + size, dat.begin() + size + N};
	}

	X prod(int l, int r) {
		assert(0 <= l && l <= r && r <= N);
		if (l == r) return MX::I();
		l += size, r += size;
		for (int i = LOG; i >= 1; i--) {
			if (((l >> i) << i) != l) push(l >> i);
			if (((r >> i) << i) != r) push((r - 1) >> i);
		}
		X xl = MX::I(), xr = MX::I();
		while (l < r) {
			if (l & 1) xl = MX::O(xl, dat[l++]);
			if (r & 1) xr = MX::O(dat[--r], xr);
			l >>= 1, r >>= 1;
		}
		return MX::O(xl, xr);
	}

	X prod() { return dat[1]; }
	template <typename... Args>
	X operator()(Args... args) { return prod(args...); }
	template <typename... Args>
	X operator[](Args... args) { return get(args...); }

	void apply(int l, int r, A a) {
		assert(0 <= l && l <= r && r <= N);
		if (l == r) return;
		l += size, r += size;
		for (int i = LOG; i >= 1; i--) {
			if (((l >> i) << i) != l) push(l >> i);
			if (((r >> i) << i) != r) push((r - 1) >> i);
		}
		int l2 = l, r2 = r;
		while (l < r) {
			if (l & 1) apply_at(l++, a);
			if (r & 1) apply_at(--r, a);
			l >>= 1, r >>= 1;
		}
		l = l2, r = r2;
		for (int i = 1; i <= LOG; i++) {
			if (((l >> i) << i) != l) update(l >> i);
			if (((r >> i) << i) != r) update((r - 1) >> i);
		}
	}

	template <typename F>
	int max_right(const F check, int l) {
		assert(0 <= l && l <= N);
		assert(check(MX::I()));
		if (l == N) return N;
		l += size;
		for (int i = LOG; i >= 1; i--) push(l >> i);
		X sm = MX::I();
		do {
			while (l % 2 == 0) l >>= 1;
			if (!check(MX::O(sm, dat[l]))) {
				while (l < size) {
					push(l);
					l = (2 * l);
					if (check(MX::O(sm, dat[l]))) {
						sm = MX::O(sm, dat[l++]);
					}
				}
				return l - size;
			}
			sm = MX::O(sm, dat[l++]);
		} while ((l & -l) != l);
		return N;
	}

	template <typename F>
	int min_left(const F check, int r) {
		assert(0 <= r && r <= N);
		assert(check(MX::I()));
		if (r == 0) return 0;
		r += size;
		for (int i = LOG; i >= 1; i--) push((r - 1) >> i);
		X sm = MX::I();
		do {
			r--;
			while (r > 1 && (r % 2)) r >>= 1;
			if (!check(MX::O(dat[r], sm))) {
				while (r < size) {
					push(r);
					r = (2 * r + 1);
					if (check(MX::O(dat[r], sm))) {
						sm = MX::O(dat[r--], sm);
					}
				}
				return r + 1 - size;
			}
			sm = MX::O(dat[r], sm);
		} while ((r & -r) != r);
		return 0;
	}

	// l <= i xor (xor_val) < r となる i 全体に apply
	void apply_xor_range(int l, int r, int xor_val, A a) {
		assert(!(N & (N - 1)));
		assert(0 <= xor_val && xor_val < N);
		assert(0 <= l && l <= r && r <= N);

		auto dfs = [&](auto& dfs, int idx, int seg_l, int seg_r) -> void {
			if (l <= seg_l && seg_r <= r) {
				return apply_at(idx, a);
			}
			if (r <= seg_l || seg_r <= l) return;
			push(idx);
			int seg_m = (seg_l + seg_r) / 2;
			int bit = (seg_r - seg_l) / 2;
			int left = 2 * idx + 0, right = 2 * idx + 1;
			if (xor_val & bit) std::swap(left, right);
			dfs(dfs, left, seg_l, seg_m);
			dfs(dfs, right, seg_m, seg_r);
			update(idx);
		};
		dfs(dfs, 1, 0, N);
	}

	private:
	inline int msb(const int &a) { return a ? 31 - __builtin_clz(a) : -1; }
	void apply_at(int k, A a) {
		long long sz = 1 << (LOG - msb(k));
		dat[k] = ActedMonoid::O(dat[k], a, sz);
		if (k < size) has_laz[k] = 1, laz[k] = MA::O(laz[k], a);
	}
	void push(int k) {
		if (!has_laz[k]) return;
		has_laz[k] = 0;
		apply_at(2 * k, laz[k]), apply_at(2 * k + 1, laz[k]);
		laz[k] = MA::I();
	}
};

template <typename T>
using sum_segtree = SegmentTree<MONOID::Add<T>>;
template <typename T>
using min_segtree = SegmentTree<MONOID::Min<T>>;
template <typename T>
using max_segtree = SegmentTree<MONOID::Max<T>>;

template <typename T>
using addmax_lazyseg = LazySegmentTree<ACTED_MONOID::Add_Max<T>>;
template <typename T>
using addmin_lazyseg = LazySegmentTree<ACTED_MONOID::Add_Min<T>>;
template <typename T>
using addsum_lazyseg = LazySegmentTree<ACTED_MONOID::Add_Sum<T>>;
template <typename T>
using updatemax_lazyseg = LazySegmentTree<ACTED_MONOID::Update_Max<T>>;
template <typename T>
using updatemin_lazyseg = LazySegmentTree<ACTED_MONOID::Update_Min<T>>;
template <typename T>
using updatesum_lazyseg = LazySegmentTree<ACTED_MONOID::Update_Sum<T>>;


#include <cassert>
#include <type_traits>
#include <vector>
using namespace std;

// コンストラクタの MAX に 「C(n, r) や fac(n) でクエリを投げる最大の n 」
// を入れると倍速くらいになる
// mod を超えて前計算して 0 割りを踏むバグは対策済み
template <typename T>
struct Binomial {
	vector<T> f, g, h;
	Binomial(int MAX = 0) {
		assert(T::get_mod() != 0 && "Binomial<mint>()");
		f.resize(1, T{1});
		g.resize(1, T{1});
		h.resize(1, T{1});
		if (MAX > 0) extend(MAX + 1);
	}

	void extend(int m = -1) {
		int n = f.size();
		if (m == -1) m = n * 2;
		m = min<int>(m, T::get_mod());
		if (n >= m) return;
		f.resize(m);
		g.resize(m);
		h.resize(m);
		for (int i = n; i < m; i++) f[i] = f[i - 1] * T(i);
		g[m - 1] = f[m - 1].inverse();
		h[m - 1] = g[m - 1] * f[m - 2];
		for (int i = m - 2; i >= n; i--) {
			g[i] = g[i + 1] * T(i + 1);
			h[i] = g[i] * f[i - 1];
		}
	}

	T fac(int i) {
		if (i < 0) return T(0);
		while (i >= (int)f.size()) extend();
		return f[i];
	}

	T finv(int i) {
		if (i < 0) return T(0);
		while (i >= (int)g.size()) extend();
		return g[i];
	}

	T inv(int i) {
		if (i < 0) return -inv(-i);
		while (i >= (int)h.size()) extend();
		return h[i];
	}

	T C(int n, int r) {
		if (n < 0 || n < r || r < 0) return T(0);
		return fac(n) * finv(n - r) * finv(r);
	}

	inline T operator()(int n, int r) { return C(n, r); }

	template <typename I>
	T multinomial(const vector<I>& r) {
		static_assert(is_integral<I>::value == true);
		int n = 0;
		for (auto& x : r) {
			if (x < 0) return T(0);
			n += x;
		}
		T res = fac(n);
		for (auto& x : r) res *= finv(x);
		return res;
	}

	template <typename I>
	T operator()(const vector<I>& r) {
		return multinomial(r);
	}

	T C_naive(int n, int r) {
		if (n < 0 || n < r || r < 0) return T(0);
		T ret = T(1);
		r = min(r, n - r);
		for (int i = 1; i <= r; ++i) ret *= inv(i) * (n--);
		return ret;
	}

	T P(int n, int r) {
		if (n < 0 || n < r || r < 0) return T(0);
		return fac(n) * finv(n - r);
	}

	// [x^r] 1 / (1-x)^n
	T H(int n, int r) {
		if (n < 0 || r < 0) return T(0);
		return r == 0 ? 1 : C(n + r - 1, r);
	}
};

template <uint32_t mod>
struct LazyMontgomeryModInt {
  using mint = LazyMontgomeryModInt;
  using i32 = int32_t;
  using u32 = uint32_t;
  using u64 = uint64_t;

  static constexpr u32 get_r() {
    u32 ret = mod;
    for (i32 i = 0; i < 4; ++i) ret *= 2 - mod * ret;
    return ret;
  }

  static constexpr u32 r = get_r();
  static constexpr u32 n2 = -u64(mod) % mod;
  static_assert(mod < (1 << 30), "invalid, mod >= 2 ^ 30");
  static_assert((mod & 1) == 1, "invalid, mod % 2 == 0");
  static_assert(r * mod == 1, "this code has bugs.");

  u32 a;

  constexpr LazyMontgomeryModInt() : a(0) {}
  constexpr LazyMontgomeryModInt(const int64_t &b)
      : a(reduce(u64(b % mod + mod) * n2)){};

  static constexpr u32 reduce(const u64 &b) {
    return (b + u64(u32(b) * u32(-r)) * mod) >> 32;
  }

  constexpr mint &operator+=(const mint &b) {
    if (i32(a += b.a - 2 * mod) < 0) a += 2 * mod;
    return *this;
  }

  constexpr mint &operator-=(const mint &b) {
    if (i32(a -= b.a) < 0) a += 2 * mod;
    return *this;
  }

  constexpr mint &operator*=(const mint &b) {
    a = reduce(u64(a) * b.a);
    return *this;
  }

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

  constexpr mint operator+(const mint &b) const { return mint(*this) += b; }
  constexpr mint operator-(const mint &b) const { return mint(*this) -= b; }
  constexpr mint operator*(const mint &b) const { return mint(*this) *= b; }
  constexpr mint operator/(const mint &b) const { return mint(*this) /= b; }
  constexpr bool operator==(const mint &b) const {
    return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a);
  }
  constexpr bool operator!=(const mint &b) const {
    return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a);
  }
  constexpr mint operator-() const { return mint() - mint(*this); }
  constexpr mint operator+() const { return mint(*this); }

  constexpr mint pow(u64 n) const {
    mint ret(1), mul(*this);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }

  constexpr mint inverse() const {
    int x = get(), y = mod, u = 1, v = 0, t = 0, tmp = 0;
    while (y > 0) {
      t = x / y;
      x -= t * y, u -= t * v;
      tmp = x, x = y, y = tmp;
      tmp = u, u = v, v = tmp;
    }
    return mint{u};
  }

  friend ostream &operator<<(ostream &os, const mint &b) {
    return os << b.get();
  }

  friend istream &operator>>(istream &is, mint &b) {
    int64_t t;
    is >> t;
    b = LazyMontgomeryModInt<mod>(t);
    return (is);
  }

  constexpr u32 get() const {
    u32 ret = reduce(a);
    return ret >= mod ? ret - mod : ret;
  }

  static constexpr u32 get_mod() { return mod; }
};
using namespace yamada;

using mint = LazyMontgomeryModInt<998244353>;
Binomial<mint> binom;
mint count_bra(ll N,ll K){ // K: count( ')' )
	if(N<K)return 0;
	return binom(N,K)-binom(N,K-1);
}

void yamada::solve()
{
	inl(N,Q);

	vc<mint> table(N+1);
	table[0]=1;
	rep(i,1,N+1)table[i]=(i%2? 1:count_bra(i,i/2));

	vc<bool> A(N);
	max_segtree<ll> seg(vl(Ceil(N,2),0)); // 食い違ったときに1
	max_segtree<ll> segmx(A);
	min_segtree<ll> segmn(A);
	mint cur=1;
	auto getL=[&](ll i){
		if(A[i])return segmn.min_left(i,[](ll mn){return mn>=1;});
		return segmx.min_left(i,[](ll mx){return mx<=0;});
	};
	auto getR=[&](ll i){
		if(A[i])return segmn.max_right(i,[](ll mn){return mn>=1;});
		return segmx.max_right(i,[](ll mx){return mx<=0;});
	};
	auto update=[&](ll i){
		// cur の更新の必要がある可能性がある区間
		ll L=getL(i);
		ll R=getR(i);
		if(i>0)amin(L,getL(i-1));
		if(i<N)amin(R,getR(i+1));

		for(ll l=L,r=L; l<R; l=r){
			r=getR(l);
			if(r<N)cur/=table[r-l];
		}
		// A,segmx,segmn
		A[i]=!A[i];
		segmx.set(i,A[i]);
		segmn.set(i,A[i]);
		{ // seg
			ll I=(i%2? i-1:i+1);
			if(I<N && A[i]!=A[I])seg.set(i/2,1);
			else seg.set(i/2,0);
		}
		// cur
		for(ll l=L,r=L; l<R; l=r){
			r=getR(l);
			if(r<N)cur*=table[r-l];
		}
	};

	ins(S);
	each(i,c,S|enumerate)if(c=='Y')update(i);


	while(Q--){
		inl(op);
		if(op==1){
			inl(i); --i;
			update(i);
		}
		else{
			inl(K);
			if(seg()){out(0); continue;}
			ll L=getL(N-1);
			ll k=K-L/2;
			if(!A[L])k=N-L-k;
			out(cur*count_bra(N-L,k));
		}
	}
}