There are N non-negative integers written on a blackboard. The i-th integer is A_i.
Takahashi can perform the following two kinds of operations any number of times in any order:
 - Select one integer written on the board (let this integer be X). Write 2X on the board, without erasing the selected integer.
 - Select two integers, possibly the same, written on the board (let these integers be X and Y). Write X XOR Y (XOR stands for bitwise xor) on the blackboard, without erasing the selected integers.
How many different integers not exceeding X can be written on the blackboard? We will also count the integers that are initially written on the board.
Since the answer can be extremely large, find the count modulo 998244353.

-----Constraints-----
 - 1 \leq N \leq 6
 - 1 \leq X < 2^{4000}
 - 1 \leq A_i < 2^{4000}(1\leq i\leq N)
 - All input values are integers.
 - X and A_i(1\leq i\leq N) are given in binary notation, with the most significant digit in each of them being 1.

-----Input-----
Input is given from Standard Input in the following format:
N X
A_1
:
A_N

-----Output-----
Print the number of different integers not exceeding X that can be written on the blackboard.

-----Sample Input-----
3 111
1111
10111
10010

-----Sample Output-----
4

Initially, 15, 23 and 18 are written on the blackboard. Among the integers not exceeding 7, four integers, 0, 3, 5 and 6, can be written.
For example, 6 can be written as follows:
 - Double 15 to write 30.
 - Take XOR of 30 and 18 to write 12.
 - Double 12 to write 24.
 - Take XOR of 30 and 24 to write 6.

Difficulty: competition

Starter code:

Solutions:
#!/usr/bin/env python3
def divmod(f, g):
    assert g
    h = 0
    for i in reversed(list(range(f.bit_length() - g.bit_length() + 1))):
        if f & (1 << (g.bit_length() + i - 1)):
            f ^= g << i
            h ^= 1 << i
    return h, f

def gcd(f, g):
    while g:
        q, r = divmod(f, g)
        f, g = g, r
    return f

import functools
def solve(n, x, a):
    # (g) = (a_1, ..., a_n) is a principal ideal since F_2[x] is a PID
    g = functools.reduce(gcd, a)

    # count h in F_2[x] s.t. h g <= x
    cnt = 0
    h = 0
    for k in reversed(list(range(x.bit_length() - g.bit_length() + 1))):
        bit = 1 << (g.bit_length() + k - 1)
        if (x & bit):
            cnt += 1 << k
        if (x & bit) != (h & bit):
            h ^= g << k
    cnt += (h <= x)
    return cnt % 998244353


def main():
    n, x = input().split()
    n = int(n)
    x = int(x, 2)
    a = [ int(input(), 2) for _ in range(n) ]
    print((solve(n, x, a)))

def __starting_point():
    main()


__starting_point()

import random

mod=998244353
N,X=input().split()
N=int(N)
A=[]
for i in range(N):
    A.append(int(input(),2))
A.sort()

a=A[-1]
M=max(len(X)-1,a.bit_length()-1)
base=[]
n=a.bit_length()-1
for i in range(M-n,-1,-1):
    base.append(a<<i)

for i in range(0,N-1):
    a=A[i]
    for j in range(M):
        for b in base:
            a=min(a,a^b)
        if a==0:
            break
        else:
            base.append(a)
            a=a<<1

data=[0]*(M+1)
data2=[0]*(M+1)
for b in base:
    data[b.bit_length()-1]=b
    data2[b.bit_length()-1]=1

for i in range(1,M+1):
    data2[i]+=data2[i-1]

data2=[0]+data2

#print(data)
#print(data2)

x=0
ans=0
n=len(X)-1
for i in range(len(X)):
    if X[i]=="1":
        if x>>(n-i)&1==1:
            if data[n-i]:
                ans+=pow(2,data2[n-i],mod)
                ans%=mod
        else:
            ans+=pow(2,data2[n-i],mod)
            ans%=mod
            if data[n-i]:
                x=x^data[n-i]
            else:
                break
    else:
        if x>>(n-i)&1==1:
            if data[n-i]:
                x=x^data[n-i]
            else:
                break
        else:
            continue
else:
    ans+=1
    ans%=mod
print(ans)


# coding: utf-8
# Your code here!
import sys
read = sys.stdin.read
readline = sys.stdin.readline
sys.setrecursionlimit(10**5)

n,x = readline().split()
n = int(n)
x = int(x,2)
*a, = [int(x,2) for x in read().split()]

def gcd(a,b):
    c = 0
    while b:
        if a < b:
            a,b = b,a
            continue
        a ^= b<<(a.bit_length() - b.bit_length())
    return a

v = 0
for ai in a: v = gcd(ai,v)

MOD = 998244353
p2 = [1]
for _ in range(5000): p2.append(p2[-1]*2%MOD)

dv = v.bit_length()
dx = x.bit_length()

def dfs(x,y,d):
    if d < dv:
        return int(x >= y)
    else:
        bx = x>>(d-1)&1
        by = y>>(d-1)&1
        if bx and by:
            return (p2[d-dv] + dfs(x,y,d-1)) %MOD
        elif not bx and by:
            return dfs(x,y^(v<<(d-dv)),d-1)
        elif not by and bx:
            return (p2[d-dv] + dfs(x,y^(v<<(d-dv)),d-1)) %MOD
        else:
            return dfs(x,y,d-1)

print((dfs(x,0,dx)))




import sys

sys.setrecursionlimit(10 ** 6)
int1 = lambda x: int(x) - 1
p2D = lambda x: print(*x, sep="\n")
def MI(): return map(int, sys.stdin.readline().split())
def LI(): return list(map(int, sys.stdin.readline().split()))
def LLI(rows_number): return [LI() for _ in range(rows_number)]

def gcd(a,b):
    if b==0:return a
    if a<b:a,b=b,a
    k=a.bit_length()-b.bit_length()
    return gcd(b,a^(b<<k))

def main():
    md=998244353
    n,x=input().split()
    x=int(x,2)
    aa=[input() for _ in range(int(n))]
    aa=[int(a,2) for a in aa]
    #bitを多項式の係数と考えて多項式のGCDを求める
    g=aa[0]
    for a in aa[1:]:
        g=gcd(g,a)
    #print(bin(g))
    #GCDの桁数以上は自由に決められるのでその分を答えに
    ans=x>>(g.bit_length()-1)
    s=0
    # GCDの桁数以上をxと同じにした場合にx以下になるようであればその分も足す
    while 1:
        k=(x^s).bit_length()-g.bit_length()
        #print(k,bin(x^s),bin(x),bin(s))
        if k<0:break
        s^=g<<k
    if s<=x:ans+=1
    print(ans%md)

main()

import random

mod=998244353
N,X=input().split()
N=int(N)
A=[]
for i in range(N):
    A.append(int(input(),2))
A.sort()

a=A[-1]
M=max(len(X)-1,a.bit_length()-1)
data=[0]*(M+1)
n=a.bit_length()-1
for i in range(M-n,-1,-1):
    data[i+n]=a<<i

low=n
for i in range(0,N-1):
    a=A[i]
    flag=True
    while flag:
        n=a.bit_length()
        for j in range(n-1,low-1,-1):
            a=min(a,a^data[j])
        if a!=0:
            data[a.bit_length()-1]=a
            id=a.bit_length()-1
            low=id
            while data[id+1]==0:
                data[id+1]=min((data[id]<<1)^a,(data[id]<<1))
                id+=1
            else:
                a=data[id]<<1
        else:
            break

data2=[0]*(M+1)
for i in range(M+1):
    data2[i]=(data[i]!=0)

for i in range(1,M+1):
    data2[i]+=data2[i-1]

data2=[0]+data2

#print(data)
#print(data2)

x=0
ans=0
n=len(X)-1
for i in range(len(X)):
    if X[i]=="1":
        if x>>(n-i)&1==1:
            if data[n-i]:
                ans+=pow(2,data2[n-i],mod)
                ans%=mod
        else:
            ans+=pow(2,data2[n-i],mod)
            ans%=mod
            if data[n-i]:
                x=x^data[n-i]
            else:
                break
    else:
        if x>>(n-i)&1==1:
            if data[n-i]:
                x=x^data[n-i]
            else:
                break
        else:
            continue
else:
    ans+=1
    ans%=mod
print(ans)


N, X = input().split()
N = int(N); X = int(X, 2)
p = int(input(), 2)
for i in range(N-1):
    v = int(input(), 2)
    if p > v:
        p, v = v, p
    while 0 < p:
        q = (v ^ (p << (v.bit_length() - p.bit_length())))
        if p < q:
            p, v = p, q
        else:
            p, v = q, p
    p = v

lx = X.bit_length()
lp = p.bit_length()

MOD = 998244353
ans = (X >> (p.bit_length() - 1)) % MOD

q = p << (lx - lp)
b = 1 << (lx - 1)

x = 0; y = 0
for k in range(lx - lp, -1, -1):
    if b & (X ^ y):
        y ^= q
    b >>= 1; q >>= 1
if y <= X:
    ans = (ans + 1) % MOD
print(ans)

import random

mod=998244353
N,X=input().split()
N=int(N)
A=[]
for i in range(N):
    A.append(int(input(),2))
A.sort()

a=A[-1]
M=max(len(X)-1,a.bit_length()-1)
data=[0]*(M+1)
n=a.bit_length()-1
for i in range(M-n,-1,-1):
    data[i+n]=a<<i

for i in range(0,N-1):
    a=A[i]
    flag=True
    while flag:
        n=a.bit_length()
        for j in range(n-1,-1,-1):
            a=min(a,a^data[j])
        if a!=0:
            data[a.bit_length()-1]=a
            id=a.bit_length()-1
            while data[id+1]==0:
                data[id+1]=min((data[id]<<1)^a,(data[id]<<1))
                if data[id+1]:
                    id+=1
                else:
                    flag=False
            else:
                a=data[id]<<1
        else:
            break

data2=[0]*(M+1)
for i in range(M+1):
    data2[i]=(data[i]!=0)

for i in range(1,M+1):
    data2[i]+=data2[i-1]

data2=[0]+data2

#print(data)
#print(data2)

x=0
ans=0
n=len(X)-1
for i in range(len(X)):
    if X[i]=="1":
        if x>>(n-i)&1==1:
            if data[n-i]:
                ans+=pow(2,data2[n-i],mod)
                ans%=mod
        else:
            ans+=pow(2,data2[n-i],mod)
            ans%=mod
            if data[n-i]:
                x=x^data[n-i]
            else:
                break
    else:
        if x>>(n-i)&1==1:
            if data[n-i]:
                x=x^data[n-i]
            else:
                break
        else:
            continue
else:
    ans+=1
    ans%=mod
print(ans)