# 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)