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 Go (1.6) (judge verdict: Accepted)

package main

import (
  "bufio"
  "fmt"
  "os"
  "strconv"
  "strings"
)

func main() {
  input := Input{bufio.NewReaderSize(os.Stdin,1000000)}
  n,m := input.NextInts()
  k := make([][]int,m)
  for i:=0;i<m;i++ { k[i] = input.NextIntArray() }
  p := input.NextIntArray()
  u := 1
  a := 0
  for i:=0;i<n;i++ { u *= 2 }
  Loop:
  for i:=0;i<u;i++ {
    s := strconv.FormatInt(int64(i),2)
    for len(s) < n { s = "0"+s }
    t := make([]int,m)
    for j:=0;j<n;j++ {
      if s[j:j+1] == "0" { continue }
      for x:=0;x<m;x++ {
        for y:=1;y<k[x][0]+1;y++ {
          if k[x][y] == j+1 { t[x]++ }
        }
      }
    }
    for x:=0;x<m;x++ {
      if t[x]%2 != p[x] { continue Loop }
    }
    a++
  }
  fmt.Println(a)
}

type Input struct {
  reader *bufio.Reader
}
func(i *Input) NextLine() string {
  var buffer []byte
  for {
    line,isPrefix,err := i.reader.ReadLine()
    if err != nil { panic(err) }
    buffer = append(buffer,line...)
    if !isPrefix { break }
  }
  return string(buffer)
}
func(i *Input) NextInts() (int,int) {
  s := strings.Split(i.NextLine()," ")
  x,_ := strconv.Atoi(s[0])
  y,_ := strconv.Atoi(s[1])
  return x,y
}
func(i *Input) NextIntArray() []int {
  s := strings.Split(i.NextLine()," ")
  a := make([]int,len(s))
  for i:=0;i<len(s);i++ { a[i],_ = strconv.Atoi(s[i]) }
  return a
}