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 Pascal (FPC 2.6.2) (judge verdict: Runtime Error)

var
	r1,r2,ans,id,N,M,i,j,k,s:Longint;
	A:array[1..10,1..10]of Boolean;
	B:array[1..10,1..11]of Boolean;
	tmp:Boolean;
begin
	read(N,M);
	for i:=1 to M do begin
		read(k);
		for j:=1 to k do begin
			read(s);
			A[i,s]:=True;
			B[i,s]:=True;
		end;
	end;
	for i:=1 to M do begin
		read(s);
		if s=1 then B[i,N+1]:=True;
	end;
	for i:=1 to M do begin
		if i>N then break;
		id:=i;
		while(id<=M)and(not A[i,id])do inc(id);
		if id>N then continue;
		for j:=i to N do begin
			tmp:=A[id,j];
			A[id,j]:=A[i,j];
			A[i,j]:=tmp;
		end;
		for j:=i+1 to M do begin
			if A[j,i]then begin
				for k:=i to N do begin
					A[j,k]:=A[j,k]xor A[i,k];
				end;
			end;
		end;
	end;
	r1:=0;
	while True do begin
		tmp:=False;
		for i:=1 to N do if A[r1+1,i]then begin
			inc(r1);
			tmp:=True;
			break;
		end;
		if(not tmp)or(r1=M)then break;
	end;
	for i:=1 to M do begin
		if i>N+1 then break;
		id:=i;
		while(id<=M)and(not B[i,id])do inc(id);
		if id>N+1 then continue;
		for j:=i to N+1 do begin
			tmp:=B[id,j];
			B[id,j]:=B[i,j];
			B[i,j]:=tmp;
		end;
		for j:=i+1 to M do begin
			if B[j,i]then begin
				for k:=i to N+1 do begin
					B[j,k]:=B[j,k]xor B[i,k];
				end;
			end;
		end;
	end;
	r2:=0;
	while True do begin
		tmp:=False;
		for i:=1 to N+1 do if B[r2+1,i]then begin
			inc(r2);
			tmp:=True;
			break;
		end;
		if(not tmp)or(r2=M)then break;
	end;
	ans:=1;
	if r1<>r2 then ans:=0;
	for i:=1 to N-r1 do ans:=ans*2;
	writeln(ans);
end.