# input import sys input = sys.stdin.readline II = lambda : int(input()) MI = lambda : map(int, input().split()) LI = lambda : [int(a) for a in input().split()] SI = lambda : input().rstrip() LLI = lambda n : [[int(a) for a in input().split()] for _ in range(n)] LSI = lambda n : [input().rstrip() for _ in range(n)] MI_1 = lambda : map(lambda x:int(x)-1, input().split()) LI_1 = lambda : [int(a)-1 for a in input().split()] def graph(n:int, m:int, dir:bool=False, index:int=-1) -> list[set[int]]: edge = [set() for i in range(n+1+index)] for _ in range(m): a,b = map(int, input().split()) a += index b += index edge[a].add(b) if not dir: edge[b].add(a) return edge def graph_w(n:int, m:int, dir:bool=False, index:int=-1) -> list[set[tuple]]: edge = [set() for i in range(n+1+index)] for _ in range(m): a,b,c = map(int, input().split()) a += index b += index edge[a].add((b,c)) if not dir: edge[b].add((a,c)) return edge mod = 998244353 inf = 1001001001001001001 ordalp = lambda s : ord(s)-65 if s.isupper() else ord(s)-97 ordallalp = lambda s : ord(s)-39 if s.isupper() else ord(s)-97 yes = lambda : print("Yes") no = lambda : print("No") yn = lambda flag : print("Yes" if flag else "No") def acc(a:list[int]): sa = [0]*(len(a)+1) for i in range(len(a)): sa[i+1] = a[i] + sa[i] return sa prinf = lambda ans : print(ans if ans < 1000001001001001001 else -1) alplow = "abcdefghijklmnopqrstuvwxyz" alpup = "ABCDEFGHIJKLMNOPQRSTUVWXYZ" alpall = "abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ" URDL = {'U':(-1,0), 'R':(0,1), 'D':(1,0), 'L':(0,-1)} DIR_4 = [[-1,0],[0,1],[1,0],[0,-1]] DIR_8 = [[-1,0],[-1,1],[0,1],[1,1],[1,0],[1,-1],[0,-1],[-1,-1]] DIR_BISHOP = [[-1,1],[1,1],[1,-1],[-1,-1]] prime60 = [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59] sys.set_int_max_str_digits(0) # sys.setrecursionlimit(10**6) # import pypyjit # pypyjit.set_param('max_unroll_recursion=-1') from collections import defaultdict from heapq import heappop,heappush from bisect import bisect_left,bisect_right DD = defaultdict BSL = bisect_left BSR = bisect_right MOD = 998244353 _IMAG = 911660635 _IIMAG = 86583718 _rate2 = (0, 911660635, 509520358, 369330050, 332049552, 983190778, 123842337, 238493703, 975955924, 603855026, 856644456, 131300601, 842657263, 730768835, 942482514, 806263778, 151565301, 510815449, 503497456, 743006876, 741047443, 56250497, 867605899, 0) _irate2 = (0, 86583718, 372528824, 373294451, 645684063, 112220581, 692852209, 155456985, 797128860, 90816748, 860285882, 927414960, 354738543, 109331171, 293255632, 535113200, 308540755, 121186627, 608385704, 438932459, 359477183, 824071951, 103369235, 0) _rate3 = (0, 372528824, 337190230, 454590761, 816400692, 578227951, 180142363, 83780245, 6597683, 70046822, 623238099, 183021267, 402682409, 631680428, 344509872, 689220186, 365017329, 774342554, 729444058, 102986190, 128751033, 395565204, 0) _irate3 = (0, 509520358, 929031873, 170256584, 839780419, 282974284, 395914482, 444904435, 72135471, 638914820, 66769500, 771127074, 985925487, 262319669, 262341272, 625870173, 768022760, 859816005, 914661783, 430819711, 272774365, 530924681, 0) def _fft(a): n = len(a) h = (n - 1).bit_length() le = 0 for le in range(0, h - 1, 2): p = 1 << (h - le - 2) rot = 1 for s in range(1 << le): rot2 = rot * rot % MOD rot3 = rot2 * rot % MOD offset = s << (h - le) for i in range(p): a0 = a[i + offset] a1 = a[i + offset + p] * rot a2 = a[i + offset + p * 2] * rot2 a3 = a[i + offset + p * 3] * rot3 a1na3imag = (a1 - a3) % MOD * _IMAG a[i + offset] = (a0 + a2 + a1 + a3) % MOD a[i + offset + p] = (a0 + a2 - a1 - a3) % MOD a[i + offset + p * 2] = (a0 - a2 + a1na3imag) % MOD a[i + offset + p * 3] = (a0 - a2 - a1na3imag) % MOD rot = rot * _rate3[(~s & -~s).bit_length()] % MOD if h - le & 1: rot = 1 for s in range(1 << (h - 1)): offset = s << 1 l = a[offset] r = a[offset + 1] * rot a[offset] = (l + r) % MOD a[offset + 1] = (l - r) % MOD rot = rot * _rate2[(~s & -~s).bit_length()] % MOD def _ifft(a): n = len(a) h = (n - 1).bit_length() le = h for le in range(h, 1, -2): p = 1 << (h - le) irot = 1 for s in range(1 << (le - 2)): irot2 = irot * irot % MOD irot3 = irot2 * irot % MOD offset = s << (h - le + 2) for i in range(p): a0 = a[i + offset] a1 = a[i + offset + p] a2 = a[i + offset + p * 2] a3 = a[i + offset + p * 3] a2na3iimag = (a2 - a3) * _IIMAG % MOD a[i + offset] = (a0 + a1 + a2 + a3) % MOD a[i + offset + p] = (a0 - a1 + a2na3iimag) * irot % MOD a[i + offset + p * 2] = (a0 + a1 - a2 - a3) * irot2 % MOD a[i + offset + p * 3] = (a0 - a1 - a2na3iimag) * irot3 % MOD irot = irot * _irate3[(~s & -~s).bit_length()] % MOD if le & 1: p = 1 << (h - 1) for i in range(p): l = a[i] r = a[i + p] a[i] = l + r if l + r < MOD else l + r - MOD a[i + p] = l - r if l - r >= 0 else l - r + MOD def ntt(a): if len(a) <= 1: return _fft(a) def intt(a): if len(a) <= 1: return _ifft(a) iv = pow(len(a), MOD - 2, MOD) for i, x in enumerate(a): a[i] = x * iv % MOD def multiply(s: list, t: list) -> list: n, m = len(s), len(t) l = n + m - 1 if min(n, m) <= 60: a = [0] * l for i, x in enumerate(s): for j, y in enumerate(t): a[i + j] += x * y return [x % MOD for x in a] z = 1 << (l - 1).bit_length() a = s + [0] * (z - n) b = t + [0] * (z - m) _fft(a) _fft(b) for i, x in enumerate(b): a[i] = a[i] * x % MOD _ifft(a) a[l:] = [] iz = pow(z, MOD - 2, MOD) return [x * iz % MOD for x in a] def pow2(s: list) -> list: n = len(s) l = (n << 1) - 1 if n <= 60: a = [0] * l for i, x in enumerate(s): for j, y in enumerate(s): a[i + j] += x * y return [x % MOD for x in a] z = 1 << (l - 1).bit_length() a = s + [0] * (z - n) _fft(a) for i, x in enumerate(a): a[i] = x * x % MOD _ifft(a) a[l:] = [] iz = pow(z, MOD - 2, MOD) return [x * iz % MOD for x in a] def ntt_doubling(a: list) -> None: M = len(a) b = a[:] intt(b) r = 1 zeta = pow(3, (MOD - 1) // (M << 1), MOD) for i, x in enumerate(b): b[i] = x * r % MOD r = r * zeta % MOD ntt(b) a += b def mod_sqrt(a: int, p: int): 'x s.t. x**2 == a (mod p) if exist else -1' if a < 2: return a if pow(a, (p - 1) >> 1, p) != 1: return -1 b = 1 while pow(b, (p - 1) >> 1, p) == 1: b += 1 m = p - 1; e = 0 while not m & 1: m >>= 1 e += 1 x = pow(a, (m - 1) >> 1, p) y = (a * x % p) * x % p x = a * x % p z = pow(b, m, p) while y != 1: j = 0 t = y while t != 1: j += 1 t = t * t % p z = pow(z, 1 << (e - j - 1), p) x = x * z % p z = z * z % p y = y * z % p e = j return x from math import log2 # https://nyaannyaan.github.io/library/fps/formal-power-series.hpp def fps_add(a: list, b: list) -> list: if len(a) < len(b): res = b[::] for i, x in enumerate(a): res[i] += x else: res = a[::] for i, x in enumerate(b): res[i] += x return [x % MOD for x in res] def fps_add_scalar(a: list, k: int) -> list: res = a[:] res[0] = (res[0] + k) % MOD return res def fps_sub(a: list, b: list) -> list: if len(a) < len(b): res = b[::] for i, x in enumerate(a): res[i] -= x res = fps_neg(res) else: res = a[::] for i, x in enumerate(b): res[i] -= x return [x % MOD for x in res] def fps_sub_scalar(a: list, k: int) -> list: return fps_add_scalar(a, -k) def fps_neg(a: list) -> list: return [MOD - x if x else 0 for x in a] def fps_mul_scalar(a: list, k: int) -> list: return [x * k % MOD for x in a] def fps_matmul(a: list, b: list) -> list: 'not verified' return [x * b[i] % MOD for i, x in enumerate(a)] def fps_div(a: list, b: list) -> list: if len(a) < len(b): return [] n = len(a) - len(b) + 1 cnt = 0 if len(b) > 64: return multiply(a[::-1][:n], fps_inv(b[::-1], n))[:n][::-1] f, g = a[::], b[::] while g and not g[-1]: g.pop() cnt += 1 coef = pow(g[-1], MOD - 2, MOD) g = fps_mul_scalar(g, coef) deg = len(f) - len(g) + 1 gs = len(g) quo = [0] * deg for i in range(deg)[::-1]: quo[i] = x = f[i + gs - 1] % MOD for j, y in enumerate(g): f[i + j] -= x * y return fps_mul_scalar(quo, coef) + [0] * cnt def fps_mod(a: list, b: list) -> list: res = fps_sub(a, multiply(fps_div(a, b), b)) while res and not res[-1]: res.pop() return res def fps_divmod(a: list, b: list): q = fps_div(a, b) r = fps_sub(a, multiply(q, b)) while r and not r[-1]: r.pop() return q, r def fps_eval(a: list, x: int) -> int: r = 0; w = 1 for v in a: r += w * v % MOD w = w * x % MOD return r % MOD def fps_inv(a: list, deg: int=-1) -> list: # assert(self[0] != 0) if deg == -1: deg = len(a) res = [0] * deg res[0] = pow(a[0], MOD - 2, MOD) d = 1 while d < deg: f = [0] * (d << 1) tmp = min(len(a), d << 1) f[:tmp] = a[:tmp] g = [0] * (d << 1) g[:d] = res[:d] ntt(f) ntt(g) for i, x in enumerate(g): f[i] = f[i] * x % MOD intt(f) f[:d] = [0] * d ntt(f) for i, x in enumerate(g): f[i] = f[i] * x % MOD intt(f) for j in range(d, min(d << 1, deg)): if f[j]: res[j] = MOD - f[j] else: res[j] = 0 d <<= 1 return res def fps_pow(a: list, k: int, deg=-1) -> list: n = len(a) if deg == -1: deg = n if k == 0: if not deg: return [] ret = [0] * deg ret[0] = 1 return ret for i, x in enumerate(a): if x: rev = pow(x, MOD - 2, MOD) ret = fps_mul_scalar(fps_exp(fps_mul_scalar(fps_log(fps_mul_scalar(a, rev)[i:], deg), k), deg), pow(x, k, MOD)) ret[:0] = [0] * (i * k) if len(ret) < deg: ret[len(ret):] = [0] * (deg - len(ret)) return ret return ret[:deg] if (i + 1) * k >= deg: break return [0] * deg def fps_exp(a: list, deg=-1) -> list: # assert(not self or self[0] == 0) if deg == -1: deg = len(a) inv = [0, 1] def inplace_integral(F: list) -> list: n = len(F) while len(inv) <= n: j, k = divmod(MOD, len(inv)) inv.append((-inv[k] * j) % MOD) return [0] + [x * inv[i + 1] % MOD for i, x in enumerate(F)] def inplace_diff(F: list) -> list: return [x * i % MOD for i, x in enumerate(F) if i] b = [1, (a[1] if 1 < len(a) else 0)] c = [1] z1 = [] z2 = [1, 1] m = 2 while m < deg: y = b + [0] * m ntt(y) z1 = z2 z = [y[i] * p % MOD for i, p in enumerate(z1)] intt(z) z[:m >> 1] = [0] * (m >> 1) ntt(z) for i, p in enumerate(z1): z[i] = z[i] * (-p) % MOD intt(z) c[m >> 1:] = z[m >> 1:] z2 = c + [0] * m ntt(z2) tmp = min(len(a), m) x = a[:tmp] + [0] * (m - tmp) x = inplace_diff(x) x.append(0) ntt(x) for i, p in enumerate(x): x[i] = y[i] * p % MOD intt(x) for i, p in enumerate(b): if not i: continue x[i - 1] -= p * i % MOD x += [0] * m for i in range(m - 1): x[m + i], x[i] = x[i], 0 ntt(x) for i, p in enumerate(z2): x[i] = x[i] * p % MOD intt(x) x.pop() x = inplace_integral(x) x[:m] = [0] * m for i in range(m, min(len(a), m << 1)): x[i] += a[i] ntt(x) for i, p in enumerate(y): x[i] = x[i] * p % MOD intt(x) b[m:] = x[m:] m <<= 1 return b[:deg] def fps_log(a: list, deg=-1) -> list: # assert(a[0] == 1) if deg == -1: deg = len(a) return fps_integral(multiply(fps_diff(a), fps_inv(a, deg))[:deg - 1]) def fps_integral(a: list) -> list: n = len(a) res = [0] * (n + 1) if n: res[1] = 1 for i in range(2, n + 1): j, k = divmod(MOD, i) res[i] = (-res[k] * j) % MOD for i, x in enumerate(a): res[i + 1] = res[i + 1] * x % MOD return res def fps_diff(a: list) -> list: return [i * x % MOD for i, x in enumerate(a) if i] def shrink(a: list) -> None: while a and not a[-1]: a.pop() class Mat: def __init__(self, a00: list, a01: list, a10: list, a11: list) -> None: self.arr = [a00, a01, a10, a11] def __mul__(self, r): a00, a01, a10, a11 = self.arr if type(r) is Mat: ra00, ra01, ra10, ra11 = r.arr A00 = fps_add(multiply(a00, ra00), multiply(a01, ra10)) A01 = fps_add(multiply(a00, ra01), multiply(a01, ra11)) A10 = fps_add(multiply(a10, ra00), multiply(a11, ra10)) A11 = fps_add(multiply(a10, ra01), multiply(a11, ra11)) shrink(A00) shrink(A01) shrink(A10) shrink(A11) return Mat(A00, A01, A10, A11) b0 = fps_add(multiply(a00, r[0]), multiply(a01, r[1])) b1 = fps_add(multiply(a10, r[0]), multiply(a11, r[1])) shrink(b0) shrink(b1) return [b0, b1] @staticmethod def I(): return Mat([1], [], [], [1]) def inner_naive_gcd(m: Mat, p: list) -> None: quo, rem = fps_divmod(p[0], p[1]) b10 = fps_sub(m.arr[0], multiply(m.arr[2], quo)) b11 = fps_sub(m.arr[1], multiply(m.arr[3], quo)) shrink(rem) shrink(b10) shrink(b11) m.arr = [m.arr[2], m.arr[3], b10, b11] p[0], p[1] = p[1], rem def inner_half_gcd(p: list) -> Mat: n = len(p[0]); m = len(p[1]) k = n + 1 >> 1 if m <= k: return Mat.I() m1 = inner_half_gcd([p[0][k:], p[1][k:]]) p = m1 * p if len(p[1]) <= k: return m1 inner_naive_gcd(m1, p) if len(p[1]) <= k: return m1 l = len(p[0]) - 1 j = 2 * k - l p[0] = p[0][j:] p[1] = p[1][j:] return inner_half_gcd(p) * m1 def inner_poly_gcd(a: list, b: list) -> Mat: p = [a[::], b[::]] shrink(p[0]); shrink(p[1]) n = len(p[0]); m = len(p[1]) if n < m: mat = inner_poly_gcd(p[1], p[0]) mat.arr = [mat.arr[1], mat.arr[0], mat.arr[2], mat.arr[3]] return mat res = Mat.I() while 1: m1 = inner_half_gcd(p) p = m1 * p if not p[1]: return m1 * res inner_naive_gcd(m1, p) if not p[1]: return m1 * res res = m1 * res def poly_gcd(a: list, b: list) -> list: p = [a, b] m = inner_poly_gcd(a, b) p = m * p if p[0]: coef = pow(p[0][-1], MOD - 2, MOD) for i, x in enumerate(p[0]): p[0][i] = x * coef % MOD return p[0] def poly_inv(f: list, g: list) -> list: p = [f, g] m = inner_poly_gcd(f, g) gcd = (m * p)[0] if len(gcd) != 1: return [0, []] x = [[1], g] return [1, fps_mul_scalar(fps_mod((m * x)[0], g), pow(gcd[0], MOD - 2, MOD))] def LinearRecurrence(n: int, p: list, q: list): """ [x^n]P(x)/Q(x) を求める deg(p) < deg(q)が必要 """ # assert len(p) < len(q) shrink(q) while n: q2 = q[:] for i in range(1,len(q2),2): q2[i] = (-q2[i])%MOD s = multiply(p,q2) t = multiply(q,q2) for i in range(n&1,len(s),2): p[i>>1] = s[i] for i in range(0,len(t),2): q[i>>1] = t[i] n >>= 1 return p[0]%MOD def Bostan_Mori(n: int, a: list, c: list): """ k 項間漸化式を求める aが初項、cが漸化式の係数 """ # assert c[0] != 0 k = len(c) if n < len(a): return a[n] c = [1] + [(-i)%MOD for i in c] p = multiply(a,c)[:k-1] return LinearRecurrence(n,p,c) n,t = MI() k,l = MI() inv6 = pow(6, -1, mod) a = [0]*t a[0] = 1 c = [0]*(t+1) c[1] = inv6*(k-1)%mod c[2] = inv6*(l-k)%mod c[t] = inv6*(6-l+1)%mod for i in range(1,t): a[i] += a[i-1] * c[1] if i >= 2: a[i] += a[i-2] * c[2] if i >= t: a[i] += a[i-t] * c[t] a[i] %= mod # print(a) res = Bostan_Mori(n-1, a, c) % mod print(res)