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 Lua (5.3.2) (judge verdict: Accepted)

local DBG = false
local function dbgpr(...)
    if DBG then
        io.write("[dbg]")
        print(...)
    end
end

local function dbgpr_t(tbl, use_pairs)
    if DBG then
        local enum = ipairs
        if use_pairs then
            enum = pairs
        end
        dbgpr(tbl)
        io.write("[dbg]")
        for i,v in enum(tbl) do
            io.write(i)
            io.write(":")
            io.write(tostring(v))
            io.write(" ")
        end
        print("")
    end
end

--

local N, M = io.read("n", "n")
local L = {}
for i=1, M do
    local k = io.read("n")
    local li = {}
    for j=1,N do
        li[j] = false
    end
    for j=1,k do
        local s = io.read("n")
        li[s] = true
    end
    table.insert(L, li)
    dbgpr("light", i)
    dbgpr_t(li)
end
local p = {}
for i=1, M do
    p[i] = io.read("n")
end
dbgpr("p")
dbgpr_t(p)
local ans = 0
for i=0,2^N-1 do
    local sw_states = {}
    for j=1,N do
        local state = (i >> (j - 1)) & 1
        sw_states[j] = state
    end
    local on = true
    for j=1,M do
        local sw_count = 0
        for k=1,N do
            if L[j][k] then
                sw_count = sw_count + sw_states[k]
            end
        end
        dbgpr("L", j, "sw_count", sw_count, "p", p[j])
        if sw_count % 2 ~= p[j] then
            on = false
        end
    end
    dbgpr_t(sw_states)
    dbgpr(on)
    if on then
        ans = ans + 1
    end
end
print(ans)