C - Switches /

### 問題文

on と off の状態を持つ N 個の スイッチと、M 個の電球があります。スイッチには 1 から N の、電球には 1 から M の番号がついています。

### 制約

• 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_i0 または 1
• 入力は全て整数である

### 入力

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


### 入力例 1

2 2
2 1 2
1 2
0 1


### 出力例 1

1

• 電球 1 は、次のスイッチのうち偶数個が on の時に点灯します: スイッチ 1, 2
• 電球 2 は、次のスイッチのうち奇数個が on の時に点灯します: スイッチ 2

(スイッチ 1、スイッチ 2) の状態としては (on,on),(on,off),(off,on),(off,off) が考えられますが、このうち (on,on) のみが条件を満たすので、1 を出力してください。

### 入力例 2

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


### 出力例 2

0

• 電球 1 は、次のスイッチのうち偶数個が on の時に点灯します: スイッチ 1, 2
• 電球 2 は、次のスイッチのうち偶数個が on の時に点灯します: スイッチ 1
• 電球 3 は、次のスイッチのうち奇数個が on の時に点灯します: スイッチ 2

よって、全ての電球が点灯するようなスイッチの on/off の状態の組み合わせは存在しないので、0 を出力してください。

### 入力例 3

5 2
3 1 2 5
2 2 3
1 0


### 出力例 3

8


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