# https://github.com/not522/ac-library-python import typing def _ceil_pow2(n: int) -> int: x = 0 while (1 << x) < n: x += 1 return x def _bsf(n: int) -> int: x = 0 while n % 2 == 0: x += 1 n //= 2 return x def _is_prime(n: int) -> bool: ''' Reference: M. Forisek and J. Jancina, Fast Primality Testing for Integers That Fit into a Machine Word ''' if n <= 1: return False if n == 2 or n == 7 or n == 61: return True if n % 2 == 0: return False d = n - 1 while d % 2 == 0: d //= 2 for a in (2, 7, 61): t = d y = pow(a, t, n) while t != n - 1 and y != 1 and y != n - 1: y = y * y % n t <<= 1 if y != n - 1 and t % 2 == 0: return False return True def _inv_gcd(a: int, b: int) -> typing.Tuple[int, int]: a %= b if a == 0: return (b, 0) # Contracts: # [1] s - m0 * a = 0 (mod b) # [2] t - m1 * a = 0 (mod b) # [3] s * |m1| + t * |m0| <= b s = b t = a m0 = 0 m1 = 1 while t: u = s // t s -= t * u m0 -= m1 * u # |m1 * u| <= |m1| * s <= b # [3]: # (s - t * u) * |m1| + t * |m0 - m1 * u| # <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u) # = s * |m1| + t * |m0| <= b s, t = t, s m0, m1 = m1, m0 # by [3]: |m0| <= b/g # by g != b: |m0| < b/g if m0 < 0: m0 += b // s return (s, m0) def _primitive_root(m: int) -> int: if m == 2: return 1 if m == 167772161: return 3 if m == 469762049: return 3 if m == 754974721: return 11 if m == 998244353: return 3 divs = [2] + [0] * 19 cnt = 1 x = (m - 1) // 2 while x % 2 == 0: x //= 2 i = 3 while i * i <= x: if x % i == 0: divs[cnt] = i cnt += 1 while x % i == 0: x //= i i += 2 if x > 1: divs[cnt] = x cnt += 1 g = 2 while True: for i in range(cnt): if pow(g, (m - 1) // divs[i], m) == 1: break else: return g g += 1 class ModContext: context: typing.List[int] = [] def __init__(self, mod: int) -> None: assert 1 <= mod self.mod = mod def __enter__(self) -> None: self.context.append(self.mod) def __exit__(self, exc_type: typing.Any, exc_value: typing.Any, traceback: typing.Any) -> None: self.context.pop() @classmethod def get_mod(cls) -> int: return cls.context[-1] class Modint: def __init__(self, v: int = 0) -> None: self._mod = ModContext.get_mod() if v == 0: self._v = 0 else: self._v = v % self._mod def mod(self) -> int: return self._mod def val(self) -> int: return self._v def __iadd__(self, rhs: typing.Union['Modint', int]) -> 'Modint': if isinstance(rhs, Modint): self._v += rhs._v else: self._v += rhs if self._v >= self._mod: self._v -= self._mod return self def __isub__(self, rhs: typing.Union['Modint', int]) -> 'Modint': if isinstance(rhs, Modint): self._v -= rhs._v else: self._v -= rhs if self._v < 0: self._v += self._mod return self def __imul__(self, rhs: typing.Union['Modint', int]) -> 'Modint': if isinstance(rhs, Modint): self._v = self._v * rhs._v % self._mod else: self._v = self._v * rhs % self._mod return self def __ifloordiv__(self, rhs: typing.Union['Modint', int]) -> 'Modint': if isinstance(rhs, Modint): inv = rhs.inv()._v else: inv = atcoder._math._inv_gcd(rhs, self._mod)[1] self._v = self._v * inv % self._mod return self def __pos__(self) -> 'Modint': return self def __neg__(self) -> 'Modint': return Modint() - self def __pow__(self, n: int) -> 'Modint': assert 0 <= n return Modint(pow(self._v, n, self._mod)) def inv(self) -> 'Modint': eg = atcoder._math._inv_gcd(self._v, self._mod) assert eg[0] == 1 return Modint(eg[1]) def __add__(self, rhs: typing.Union['Modint', int]) -> 'Modint': if isinstance(rhs, Modint): result = self._v + rhs._v if result >= self._mod: result -= self._mod return raw(result) else: return Modint(self._v + rhs) def __sub__(self, rhs: typing.Union['Modint', int]) -> 'Modint': if isinstance(rhs, Modint): result = self._v - rhs._v if result < 0: result += self._mod return raw(result) else: return Modint(self._v - rhs) def __mul__(self, rhs: typing.Union['Modint', int]) -> 'Modint': if isinstance(rhs, Modint): return Modint(self._v * rhs._v) else: return Modint(self._v * rhs) def __floordiv__(self, rhs: typing.Union['Modint', int]) -> 'Modint': if isinstance(rhs, Modint): inv = rhs.inv()._v else: inv = atcoder._math._inv_gcd(rhs, self._mod)[1] return Modint(self._v * inv) def __eq__(self, rhs: typing.Union['Modint', int]) -> bool: # type: ignore if isinstance(rhs, Modint): return self._v == rhs._v else: return self._v == rhs def __ne__(self, rhs: typing.Union['Modint', int]) -> bool: # type: ignore if isinstance(rhs, Modint): return self._v != rhs._v else: return self._v != rhs def raw(v: int) -> Modint: x = Modint() x._v = v return x _sum_e = {} # _sum_e[i] = ies[0] * ... * ies[i - 1] * es[i] def _butterfly(a: typing.List[Modint]) -> None: g = atcoder._math._primitive_root(a[0].mod()) n = len(a) h = atcoder._bit._ceil_pow2(n) if a[0].mod() not in _sum_e: es = [Modint(0)] * 30 # es[i]^(2^(2+i)) == 1 ies = [Modint(0)] * 30 cnt2 = atcoder._bit._bsf(a[0].mod() - 1) e = Modint(g) ** ((a[0].mod() - 1) >> cnt2) ie = e.inv() for i in range(cnt2, 1, -1): # e^(2^i) == 1 es[i - 2] = e ies[i - 2] = ie e = e * e ie = ie * ie sum_e = [Modint(0)] * 30 now = Modint(1) for i in range(cnt2 - 2): sum_e[i] = es[i] * now now *= ies[i] _sum_e[a[0].mod()] = sum_e else: sum_e = _sum_e[a[0].mod()] for ph in range(1, h + 1): w = 1 << (ph - 1) p = 1 << (h - ph) now = Modint(1) for s in range(w): offset = s << (h - ph + 1) for i in range(p): left = a[i + offset] right = a[i + offset + p] * now a[i + offset] = left + right a[i + offset + p] = left - right now *= sum_e[atcoder._bit._bsf(~s)] _sum_ie = {} # _sum_ie[i] = es[0] * ... * es[i - 1] * ies[i] def _butterfly_inv(a: typing.List[Modint]) -> None: g = atcoder._math._primitive_root(a[0].mod()) n = len(a) h = atcoder._bit._ceil_pow2(n) if a[0].mod() not in _sum_ie: es = [Modint(0)] * 30 # es[i]^(2^(2+i)) == 1 ies = [Modint(0)] * 30 cnt2 = atcoder._bit._bsf(a[0].mod() - 1) e = Modint(g) ** ((a[0].mod() - 1) >> cnt2) ie = e.inv() for i in range(cnt2, 1, -1): # e^(2^i) == 1 es[i - 2] = e ies[i - 2] = ie e = e * e ie = ie * ie sum_ie = [Modint(0)] * 30 now = Modint(1) for i in range(cnt2 - 2): sum_ie[i] = ies[i] * now now *= es[i] _sum_ie[a[0].mod()] = sum_ie else: sum_ie = _sum_ie[a[0].mod()] for ph in range(h, 0, -1): w = 1 << (ph - 1) p = 1 << (h - ph) inow = Modint(1) for s in range(w): offset = s << (h - ph + 1) for i in range(p): left = a[i + offset] right = a[i + offset + p] a[i + offset] = left + right a[i + offset + p] = Modint( (a[0].mod() + left.val() - right.val()) * inow.val()) inow *= sum_ie[atcoder._bit._bsf(~s)] def convolution_mod(a: typing.List[Modint], b: typing.List[Modint]) -> typing.List[Modint]: n = len(a) m = len(b) if n == 0 or m == 0: return [] if min(n, m) <= 60: if n < m: n, m = m, n a, b = b, a ans = [Modint(0) for _ in range(n + m - 1)] for i in range(n): for j in range(m): ans[i + j] += a[i] * b[j] return ans z = 1 << atcoder._bit._ceil_pow2(n + m - 1) while len(a) < z: a.append(Modint(0)) _butterfly(a) while len(b) < z: b.append(Modint(0)) _butterfly(b) for i in range(z): a[i] *= b[i] _butterfly_inv(a) a = a[:n + m - 1] iz = Modint(z).inv() for i in range(n + m - 1): a[i] *= iz return a def convolution(mod: int, a: typing.List[typing.Any], b: typing.List[typing.Any]) -> typing.List[typing.Any]: n = len(a) m = len(b) if n == 0 or m == 0: return [] with ModContext(mod): a2 = list(map(Modint, a)) b2 = list(map(Modint, b)) return list(map(lambda c: c.val(), convolution_mod(a2, b2))) def convolution_int( a: typing.List[int], b: typing.List[int]) -> typing.List[int]: n = len(a) m = len(b) if n == 0 or m == 0: return [] mod1 = 754974721 # 2^24 mod2 = 167772161 # 2^25 mod3 = 469762049 # 2^26 m2m3 = mod2 * mod3 m1m3 = mod1 * mod3 m1m2 = mod1 * mod2 m1m2m3 = mod1 * mod2 * mod3 i1 = atcoder._math._inv_gcd(mod2 * mod3, mod1)[1] i2 = atcoder._math._inv_gcd(mod1 * mod3, mod2)[1] i3 = atcoder._math._inv_gcd(mod1 * mod2, mod3)[1] c1 = convolution(mod1, a, b) c2 = convolution(mod2, a, b) c3 = convolution(mod3, a, b) c = [0] * (n + m - 1) for i in range(n + m - 1): c[i] += (c1[i] * i1) % mod1 * m2m3 c[i] += (c2[i] * i2) % mod2 * m1m3 c[i] += (c3[i] * i3) % mod3 * m1m2 c[i] %= m1m2m3 return c from collections import deque n, m = map(int, input().split()) a = list(map(int, input().split())) s = input() MOD = 998244353 # g_j(x) を (1-x)^(e_1) (1-2x)^(e_2) (1-3x)^(e_3) ... と表したときの指数列を更新していく. q = deque() q.append(-1) q_sum = -1 for i in range(m): if s[i] == 's': q.appendleft(-q_sum - 1) q_sum = -1 elif s[i] == 'a': q[0] -= 1 q_sum -= 1 q.appendleft(0) K = len(q) # g_M(x) の (1-kx) たちを分子と分母に振り分ける. nums = [[1]] dnms = [[1]] for k in range(1, K): if q[k] > 0: for _ in range(q[k]): nums.append([1, -k]) elif q[k] < 0: for _ in range(-q[k]): dnms.append([1, -k]) # 分子を分割統治法で求める. Dnum = len(nums) d = 1 while (d < Dnum): for i in range(0, Dnum - d, 2 * d): nums[i] = convolution(MOD, nums[i], nums[i + d]) d <<= 1 # 分母を分割統治法で求める. Ddnm = len(dnms) d = 1 while (d < Ddnm): for i in range(0, Ddnm - d, 2 * d): dnms[i] = convolution(MOD, dnms[i], dnms[i + d]) d <<= 1 # 分母の形式的冪級数としての逆元を求める. while len(dnms[0]) > n: dnms[0].pop() while len(dnms[0]) < n: dnms[0].append(0) dnm_inv = [1] k = 1 while (k < n): l = min(2 * k, n) tmp = [0] * l i_ub = min(l, n) for i in range(i_ub): tmp[i] = -dnms[0][i] tmp = convolution(MOD, tmp, dnm_inv) while len(tmp) > l: tmp.pop() tmp[0] += 2 dnm_inv = convolution(MOD, tmp, dnm_inv) while len(dnm_inv) > l: dnm_inv.pop() k <<= 1 # g_M(x) を求める. f = convolution(MOD, nums[0], dnm_inv) # 答えへの寄与を足し合わせる. res = 0 for i in range(n): l = i r = n - 1 - i res = (res + a[i] * f[l] % MOD * f[r] % MOD) % MOD print(res)