Problem:
Score : 1000 points

Problem Statement

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 1

3 111
1111
10111
10010

Sample Output 1

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.

Sample Input 2

4 100100
1011
1110
110101
1010110

Sample Output 2

37

Sample Input 3

4 111001100101001
10111110
1001000110
100000101
11110000011

Sample Output 3

1843

Sample Input 4

1 111111111111111111111111111111111111111111111111111111111111111
1

Sample Output 4

466025955

Be sure to find the count modulo 998244353.

Solution in Rust (1.15.1) (judge verdict: Accepted)

#[allow(unused_imports)]
use std::cmp::*;
#[allow(unused_imports)]
use std::collections::*;
use std::io::{Write, BufWriter};
// https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8
macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        let mut next = || { iter.next().unwrap() };
        input_inner!{next, $($r)*}
    };
    ($($r:tt)*) => {
        let stdin = std::io::stdin();
        let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));
        let mut next = move || -> String{
            bytes
                .by_ref()
                .map(|r|r.unwrap() as char)
                .skip_while(|c|c.is_whitespace())
                .take_while(|c|!c.is_whitespace())
                .collect()
        };
        input_inner!{next, $($r)*}
    };
}

macro_rules! input_inner {
    ($next:expr) => {};
    ($next:expr, ) => {};

    ($next:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };
}

macro_rules! read_value {
    ($next:expr, ( $($t:tt),* )) => {
        ( $(read_value!($next, $t)),* )
    };

    ($next:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    };

    ($next:expr, chars) => {
        read_value!($next, String).chars().collect::<Vec<char>>()
    };

    ($next:expr, usize1) => {
        read_value!($next, usize) - 1
    };

    ($next:expr, [ $t:tt ]) => {{
        let len = read_value!($next, usize);
        (0..len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    }};

    ($next:expr, $t:ty) => {
        $next().parse::<$t>().expect("Parse error")
    };
}

/// Verified by: https://atcoder.jp/contests/pakencamp-2018-day3/submissions/3878249
mod mod_int {
    use std::ops::*;
    pub trait Mod: Copy + Clone { fn m() -> i64; }
    #[derive(Copy, Clone, Hash, PartialEq, Eq, PartialOrd, Ord)]
    pub struct ModInt<M> { pub x: i64, phantom: ::std::marker::PhantomData<*const M> }
    impl<M: Mod> ModInt<M> {
        fn check_integrity(self) {
            debug_assert!(self.x >= 0);
            debug_assert!(self.x < M::m());
        }
        // x >= 0
        pub fn new(x: i64) -> Self { ModInt::new_internal(x % M::m()) }
        fn new_internal(x: i64) -> Self {
            ModInt { x: x, phantom: ::std::marker::PhantomData }
        }
        #[allow(dead_code)]
        pub fn mul_fast(self, other: Self) -> Self {
            self.check_integrity();
            other.check_integrity();
            ModInt::new_internal(self.x * other.x % M::m())
        }
        #[allow(dead_code)]
        pub fn mul_slow(self, other: Self) -> Self {
            // Naive multiplication in order to avoid overflow
            self.check_integrity();
            other.check_integrity();
            let mut sum = ModInt::new_internal(0);
            let mut cur = self;
            let mut e = other.x;
            if self.x < other.x {
                cur = other;
                e = self.x;
            }
            while e > 0 {
                if e % 2 == 1 { sum += cur; }
                cur += cur;
                e /= 2;
            }
            sum
        }
        pub fn pow(self, mut e: i64) -> Self {
            self.check_integrity();
            debug_assert!(e >= 0);
            let mut sum = ModInt::new_internal(1);
            let mut cur = self;
            while e > 0 {
                if e % 2 != 0 { sum *= cur; }
                cur *= cur;
                e /= 2;
            }
            sum
        }
        #[allow(dead_code)]
        pub fn inv(self) -> Self { self.pow(M::m() - 2) }
    }
    impl<M: Mod> Add for ModInt<M> {
        type Output = Self;
        fn add(self, other: Self) -> Self {
            self.check_integrity();
            other.check_integrity();
            let mut sum = self.x + other.x;
            if sum >= M::m() { sum -= M::m(); }
            ModInt::new_internal(sum)
        }
    }
    impl<M: Mod> Sub for ModInt<M> {
        type Output = Self;
        fn sub(self, other: Self) -> Self {
            self.check_integrity();
            other.check_integrity();
            let mut sum = self.x - other.x;
            if sum < 0 { sum += M::m(); }
            ModInt::new_internal(sum)
        }
    }
    impl<M: Mod> Mul for ModInt<M> {
        type Output = Self;
        fn mul(self, other: Self) -> Self { self.mul_fast(other) }
    }
    impl<M: Mod> AddAssign for ModInt<M> {
        fn add_assign(&mut self, other: Self) { *self = *self + other; }
    }
    impl<M: Mod> SubAssign for ModInt<M> {
        fn sub_assign(&mut self, other: Self) { *self = *self - other; }
    }
    impl<M: Mod> MulAssign for ModInt<M> {
        fn mul_assign(&mut self, other: Self) { *self = *self * other; }
    }
    impl<M> ::std::fmt::Display for ModInt<M> {
        fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
            self.x.fmt(f)
        }
    }
    impl<M> ::std::fmt::Debug for ModInt<M> {
        fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
            self.x.fmt(f)
        }
    }
} // mod mod_int

macro_rules! define_mod {
    ($struct_name: ident, $modulo: expr) => {
        #[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Hash)]
        struct $struct_name {}
        impl mod_int::Mod for $struct_name { fn m() -> i64 { $modulo } }
    }
}
const MOD: i64 = 998_244_353;
define_mod!(P, MOD);
type ModInt = mod_int::ModInt<P>;

#[derive(Clone)]
struct BitSet {
    size: usize,
    buf: Vec<usize>,
}

impl BitSet {
    // size should be a multiple of bit-size of usize.
    fn new(size: usize) -> Self {
        let w = 8 * std::mem::size_of::<usize>();
        assert_eq!(size & (w - 1), 0);
        let count = size / w;
        BitSet {
            size: size,
            buf: vec![0; count],
        }
    }
    fn set(&mut self, idx: usize, val: bool) {
        debug_assert!(idx < self.size);
        let w = 8 * std::mem::size_of::<usize>();
        let idx0 = idx / w;
        let idx1 = idx & (w - 1);
        if val {
            self.buf[idx0] |= 1 << idx1;
        } else {
            self.buf[idx0] &= !(1 << idx1);
        }
    }
    fn shl(&self, val: usize) -> Self {
        if val >= self.size { return Self::new(self.size); }
        let w = 8 * std::mem::size_of::<usize>();
        let count = self.size / w;
        let sh0 = val / w;
        let sh1 = val & (w - 1);
        let mut ans = Self::new(self.size);
        if sh1 == 0 {
            for i in 0 .. count - sh0 {
                ans.buf[i + sh0] = self.buf[i];
            }
        } else {
            ans.buf[sh0] = self.buf[0] << sh1;
            for i in 1 .. count - sh0 {
                ans.buf[i + sh0] = self.buf[i] << sh1
                    | self.buf[i - 1] >> (w - sh1);
            }
        }
        ans
    }
    fn shr(&self, val: usize) -> Self {
        if val >= self.size { return Self::new(self.size); }
        let w = 8 * std::mem::size_of::<usize>();
        let count = self.size / w;
        let sh0 = val / w;
        let sh1 = val & (w - 1);
        let mut ans = Self::new(self.size);
        if sh1 == 0 {
            for i in 0 .. count - sh0 {
                ans.buf[i] = self.buf[i + sh0];
            }
        } else {
            for i in 0 .. count - sh0 - 1 {
                ans.buf[i] = self.buf[i + sh0] >> sh1
                    | self.buf[i + sh0 + 1] << (w - sh1);
            }
            ans.buf[self.size - sh0 - 1] = self.buf[self.size - 1] >> sh1;
        }
        ans
    }
    fn msb(&self) -> Option<usize> {
        let w = 8 * std::mem::size_of::<usize>();
        let count = self.size / w;
        for i in (0 .. count).rev() {
            if self.buf[i] != 0 {
                return Some(w * i + w - 1 - self.buf[i].leading_zeros() as usize);
            }
        }
        None
    }
    fn get(&self, idx: usize) -> bool {
        let w = 8 * std::mem::size_of::<usize>();
        debug_assert!(idx < self.size);
        let idx0 = idx / w;
        let idx1 = idx & (w - 1);
        (self.buf[idx0] >> idx1 & 1) == 1
    }
}
impl std::fmt::Display for BitSet {
    fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
        let size = match self.msb() {
            None => return f.write_str("0"),
            Some(size) => size + 1,
        };
        for i in (0 .. size).rev() {
            f.write_str(if self.get(i) { "1" } else { "0" })?
        }
        Ok(())
    }
}
impl std::fmt::Debug for BitSet {
    fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
        std::fmt::Display::fmt(&self, f)
    }
}

// TODO reference is not allowed as rhs
impl<'a> std::ops::BitXorAssign for BitSet {
    fn bitxor_assign(&mut self, other: BitSet) {
        debug_assert_eq!(self.size, other.size);
        let w = 8 * std::mem::size_of::<usize>();
        let count = self.size / w;
        for i in 0 .. count {
            self.buf[i] ^= other.buf[i];
        }
    }
}

fn gcd(mut a: BitSet, mut b: BitSet) -> BitSet {
    let (mut da, mut db) = match (a.msb(), b.msb()) {
        (None, _) => return b,
        (_, None) => return a,
        (Some(da), Some(db)) => (da, db),
    };
    if da < db {
        std::mem::swap(&mut da, &mut db);
        std::mem::swap(&mut a, &mut b);
    }

    for i in (db .. da + 1).rev() {
        if a.get(i) {
            a ^= b.shl(i - db);
        }
    }
    gcd(b, a)
}

const W: usize = 4032;
fn to_bitset(mut c: Vec<char>) -> BitSet {
    let mut ans = BitSet::new(W);
    c.reverse();
    for i in 0 .. c.len() {
        if c[i] == '1' {
            ans.set(i, true);
        }
    }
    ans
}

fn solve() {
    let out = std::io::stdout();
    let mut out = BufWriter::new(out.lock());
    macro_rules! puts {
        ($($format:tt)*) => (write!(out,$($format)*).unwrap());
    }
    input! {
        n: usize,
        x: chars,
        a: [chars; n],
    }
    let x = to_bitset(x);
    let a: Vec<BitSet> = a.into_iter().map(to_bitset).collect();
    let mut g = to_bitset(vec!['0']);
    for a in a { g = gcd(g, a); }
    let hi = g.msb().unwrap();
    let mut y = to_bitset(vec!['0']);
    let mut r = x.clone();
    // What is the largest multiple of g that is <= x?
    for i in (hi .. W).rev() {
        if r.get(i) {
            r ^= g.shl(i - hi);
            y.set(i - hi, true);
        }
    }
    let mut z = x.clone();
    z ^= r;
    let mut lt = false;
    for i in (0 .. W).rev() {
        if x.get(i) != z.get(i) {
            lt = x.get(i) < z.get(i);
            break;
        }
    }
    let mut tot = ModInt::new(0);
    for i in (hi .. W).rev() {
        tot += tot;
        if z.get(i) {
            tot += ModInt::new(1);
        }
    }
    if !lt {
        tot += ModInt::new(1);
    }
    puts!("{}\n", tot);
}

fn main() {
    // In order to avoid potential stack overflow, spawn a new thread.
    let stack_size = 104_857_600; // 100 MB
    let thd = std::thread::Builder::new().stack_size(stack_size);
    thd.spawn(|| solve()).unwrap().join().unwrap();
}