p02563 AtCoder Library Practice Contest - Convolution
You are given two integer arrays a_0, a_1, ..., a_{N - 1} and b_0, b_1, ..., b_{M - 1}. Calculate the array c_0, c_1, ..., c_{(N - 1) + (M - 1)}, defined by c_i = \sum_{j = 0}^i a_j b_{i - j} \bmod 998244353.

Constraints

* 1 \leq N, M \leq 524288
* 0 \leq a_i, b_i < 998244353
* All values in Input are integer.

Input

Input is given from Standard Input in the following format:


N M
a_0 a_1 ... a_{N-1}
b_0 b_1 ... b_{M-1}


Output

Print the answer in the following format:


c_0 c_1 ... c_{(N - 1) + (M - 1)}

Output

Print the answer in the following format:


c_0 c_1 ... c_{(N - 1) + (M - 1)}

Examples

Input

4 5
1 2 3 4
5 6 7 8 9


Output

5 16 34 60 70 70 59 36


Input

1 1
10000000
10000000


Output

871938225
```python
#Convolution_998244353

MOD = 998244353
ROOT = 3

sum_e = (911660635, 509520358, 369330050, 332049552, 983190778, 123842337, 238493703, 975955924, 603855026, 856644456, 131300601, 842657263, 730768835, 942482514, 806263778, 151565301, 510815449, 503497456, 743006876, 741047443, 56250497, 0, 0, 0, 0, 0, 0, 0, 0, 0)
sum_ie = (86583718, 372528824, 373294451, 645684063, 112220581, 692852209, 155456985, 797128860, 90816748, 860285882, 927414960, 354738543, 109331171, 293255632, 535113200, 308540755, 121186627, 608385704, 438932459, 359477183, 824071951, 0, 0, 0, 0, 0, 0, 0, 0, 0)

def butterfly(arr):
    n = len(arr)
    h = (n - 1).bit_length()
    for ph in range(1, h + 1):
        w = 1 << (ph - 1)
        p = 1 << (h - ph)
        now = 1
        for s in range(w):
            offset = s << (h - ph + 1)
            for i in range(p):
                l = arr[i + offset]
                r = arr[i + offset + p] * now
                arr[i + offset] = (l + r) % MOD
                arr[i + offset + p] = (l - r) % MOD
            now *= sum_e[(~s & -~s).bit_length() - 1]
            now %= MOD

def butterfly_inv(arr):
    n = len(arr)
    h = (n - 1).bit_length()
    for ph in range(1, h + 1)[::-1]:
        w = 1 << (ph - 1)
        p = 1 << (h - ph)
        inow = 1
        for s in range(w):
            offset = s << (h - ph + 1)
            for i in range(p):
                l = arr[i + offset]
                r = arr[i + offset + p]
                arr[i + offset] = (l + r) % MOD
                arr[i + offset + p] = (MOD + l - r) * inow % MOD
            inow *= sum_ie[(~s & -~s).bit_length() - 1]
            inow %= MOD

def convolution(a, b):
    n = len(a)
    m = len(b)
    if not n or not m: return []
    if min(n, m) <= 50:
        if n < m:
            n, m = m, n
            a, b = b, a
        res = [0] * (n + m - 1)
        for i in range(n):
            for j in range(m):
                res[i + j] += a[i] * b[j]
                res[i + j] %= MOD
        return res
    z = 1 << (n + m - 2).bit_length()
    a += [0] * (z - n)
    b += [0] * (z - m)
    butterfly(a)
    butterfly(b)
    for i in range(z):
        a[i] *= b[i]
        a[i] %= MOD
    butterfly_inv(a)
    a = a[:n + m - 1]
    iz = pow(z, MOD - 2, MOD)
    for i in range(n + m - 1):
        a[i] *= iz
        a[i] %= MOD
    return a

def autocorrelation(a):
    n = len(a)
    if not n: return []
    if n <= 50:
        res = [0] * (2 * n - 1)
        for i in range(n):
            for j in range(n):
                res[i + j] += a[i] * a[j]
                res[i + j] %= MOD
        return res
    z = 1 << (2 * n - 2).bit_length()
    a += [0] * (z - n)
    butterfly(a)
    for i in range(a):
        a[i] *= a[i]
        a[i] %= MOD
    butterfly_inv(a)
    a = a[:2 * n - 1]
    iz = pow(z, MOD - 2, MOD)
    for i in range(2 * n - 1):
        a[i] *= iz
        a[i] %= MOD
    return a

N, M = map(int, input().split())
A = [int(x) for x in input().split()]
B = [int(x) for x in input().split()]

print(*convolution(A, B))
```