Problem:
Score : 300 points

Problem Statement

We have N switches with "on" and "off" state, and M bulbs. The switches are numbered 1 to N, and the bulbs are numbered 1 to M.

Bulb i is connected to k_i switches: Switch s_{i1}, s_{i2}, ..., and s_{ik_i}. It is lighted when the number of switches that are "on" among these switches is congruent to p_i modulo 2.

How many combinations of "on" and "off" states of the switches light all the bulbs?

Constraints

1 \leq N, M \leq 10

1 \leq k_i \leq N

1 \leq s_{ij} \leq N

s_{ia} \neq s_{ib} (a \neq b)

p_i is 0 or 1.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N M
k_1 s_{11} s_{12} ... s_{1k_1}
:
k_M s_{M1} s_{M2} ... s_{Mk_M}
p_1 p_2 ... p_M

Output

Print the number of combinations of "on" and "off" states of the switches that light all the bulbs.

Sample Input 1

2 2
2 1 2
1 2
0 1

Sample Output 1

1

Bulb 1 is lighted when there is an even number of switches that are "on" among the following: Switch 1 and 2.

Bulb 2 is lighted when there is an odd number of switches that are "on" among the following: Switch 2.

There are four possible combinations of states of (Switch 1, Switch 2): (on, on), (on, off), (off, on) and (off, off). Among them, only (on, on) lights all the bulbs, so we should print 1.

Sample Input 2

2 3
2 1 2
1 1
1 2
0 0 1

Sample Output 2

0

Bulb 1 is lighted when there is an even number of switches that are "on" among the following: Switch 1 and 2.

Bulb 2 is lighted when there is an even number of switches that are "on" among the following: Switch 1.

Bulb 3 is lighted when there is an odd number of switches that are "on" among the following: Switch 2.

Switch 1 has to be "off" to light Bulb 2 and Switch 2 has to be "on" to light Bulb 3, but then Bulb 1 will not be lighted. Thus, there are no combinations of states of the switches that light all the bulbs, so we should print 0.

Sample Input 3

5 2
3 1 2 5
2 2 3
1 0

Sample Output 3

8

Solution in C (GCC 5.4.1) (judge verdict: Accepted)

#include <stdio.h>
#include <string.h>
#include <stdlib.h>

#define BUF_SIZE 40
#define MAX_SIZE 11
#define DIVISOR 2

int get_int2(int *a1, int *a2) {
#ifdef BUF_SIZE
  char line[BUF_SIZE];
  if(!fgets(line, BUF_SIZE, stdin)) return -1;
  sscanf(line, "%d %d", a1, a2);
#else
#error
#endif
  return 0;
}

// <size of arr> <arr[0]> <arr[1]> .. <arr[n-1]>
int fget_array_with_size(int *arr) {
#ifdef BUF_SIZE
    char line[BUF_SIZE];
    char *tmpbuf = line;
    int i;
    if(!fgets(line, BUF_SIZE, stdin)) return -1;
    char *tmp = strtok(tmpbuf, " ");
    arr[0] = strtol(tmp, NULL, 10);
    tmpbuf = NULL;
    for(i = 1; i <= arr[0]; i++) {
        char *tmp = strtok(tmpbuf, " ");
        arr[i] = strtol(tmp, NULL, 10);
    }
#else
#error
#endif
    return 0;
}

// <arr[0]> <arr[1]> .. <arr[size-1]>
int fget_array(int *arr, int size) {
#ifdef BUF_SIZE
    char line[BUF_SIZE];
    char *tmpbuf = line;
    int i;
    if(!fgets(line, BUF_SIZE, stdin)) return -1;
    for(i = 0; i < size; i++) {
        char *tmp = strtok(tmpbuf, " ");
        arr[i] = strtol(tmp, NULL, 10);
        tmpbuf = NULL;
    }
#else
#error
#endif
    return 0;
}

int main(void) {
    int switches, lights;
    get_int2(&switches, &lights);
    int switch_bits[MAX_SIZE] = {0};
    int arr[MAX_SIZE];
    int i, j;
    for(i = 0; i < lights; i++) {
        fget_array_with_size(arr);
        for(j = 1; j <= arr[0]; j++) {
            switch_bits[i] |= (1<<(arr[j]-1));
        }
    }
    fget_array(arr, lights);

    int ans = 0;
    int pattern;
    for(pattern = 0; pattern < (1<<switches); pattern++) {
        int flag = 1;
        for(j = 0; j < lights; j++) {
            int cnt = __builtin_popcount(pattern & switch_bits[j]);
            //printf("light %d -> %d <-> %d\n", j, cnt, arr[j]);
            if(cnt % DIVISOR != arr[j]) {
                flag = 0;
                break;
            }
        }
        ans += flag;
    }
    printf("%d\n", ans);
    return 0;
}