
Score : 300 points
Problem StatementWe 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.
InputInput 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
OutputPrint the number of combinations of "on" and "off" states of the switches that light all the bulbs.
Sample Input 12 2
2 1 2
1 2
0 1
Sample Output 11
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 22 3
2 1 2
1 1
1 2
0 0 1
Sample Output 20
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 35 2
3 1 2 5
2 2 3
1 0
Sample Output 38
