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 JavaScript (node.js v5.12) (judge verdict: Accepted)

// inputに入力データ全体が入る
function Main(input) {
    // 1行目がinput[0], 2行目がinput[1], …に入る
    input = input.split("\n");
    tmp0 = input[0].split(" ");
    N = Number(tmp0[0]);
    M = Number(tmp0[1])

    var s = {}
    for(var i=1; i <= M; i++)
    {
        var tmp = input[i].split(" ")
        var k = Number(tmp[0])
        var val = 0
        for(var j=1; j <= k; j++)
        {
            val += Math.pow(2,Number(tmp[j])-1)
        }
        // console.log((Array(N).join('0') + val.toString(2)).slice(-N))
        lamp_switch = val
        s[i] = lamp_switch
    }
    var tmpp = input[M+1].split(" ")
    
    var ans =0
    for(var j=0; j <Math.pow(2,N); j++)
    {
        // console.log((Array(N).join('0') + j.toString(2)).slice(-N))
        var switch_states = j
        var flashCnt=0
        for(var i=1; i <= M; i++)
        {
            //電球i
            var andStr = (Array(N).join('0') + (switch_states & s[i]).toString(2)).slice(-N)
            var onCnt = andStr.split("1").length-1
            // console.log(`電球${i}`)
            // console.log("lamp_switch: " + (Array(N).join('0') + s[i].toString(2)).slice(-N))
            // console.log("switch_states: " + (Array(N).join('0') + j.toString(2)).slice(-N))
            // console.log("and: " + andStr)

            if((onCnt%2 == Number(tmpp[i-1])))
            {
                flashCnt++
            }
        }
        if(flashCnt == M){
            ans++
        }
    }
    console.log(ans)
  }
  //*この行以降は編集しないでください（標準入出力から一度に読み込み、Mainを呼び出します）
  Main(require("fs").readFileSync("/dev/stdin", "utf8"));
//   Main("5 2\n3 1 2 5\n2 2 3\n1 0\n");