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 OCaml (4.02.3) (judge verdict: Accepted)

open Batteries
module EnumL = Enum.Labels
open Tuple

let dbg x = Printf.eprintf "[debug]%s\n" @@ dump x; x

let scan fmt f =
  Scanf.sscanf (read_line ()) fmt f

let scan_lines n fmt f =
  List.Labels.map
    (List.range 1 `To n)
    ~f:(fun _ -> scan fmt f)
  |> List.enum

let scan_list cnv =
  let l = read_line ()
          |> String.split_on_char ' '
          |> List.map cnv in
  List.enum l

let scan_lists n cnv =
  let ll = List.Labels.map (List.range 1 `To n)
      ~f:(fun _ -> scan_list cnv) in
  List.enum ll

let scan_array n m e conv =
  let arr = Array.make_matrix n m e in
  EnumL.iter (0 --^ n)
    ~f:(fun i -> arr.(i) <- Array.of_enum @@ scan_list conv);
  arr

let (n, m) = scan "%d %d" Tuple2.make
let lights = scan_lists m Int.of_string
let onoff = scan_list Int.of_string

let rec superset e =
  match Enum.get e with
  | None -> List.enum [Enum.empty ()]
  | Some v ->
    let f = superset e in
    let g = Enum.clone f in
    EnumL.map f ~f:(fun f -> let h = Enum.clone f in push h v; h)
    |>  Enum.append g

let intersection l =
  EnumL.filter ~f:(fun x -> EnumL.exists ~f:((=) x) @@ Enum.clone l)

let () =
  let sets = superset (1 -- n) in
  EnumL.map sets ~f:(fun set ->
      (* EnumL.map (Enum.combine (Enum.clone onoff) (Enum.clone lights)) *)
      EnumL.map (Enum.combine (Enum.clone onoff, Enum.clone lights))
        ~f:(fun (b, l) ->
            let n = Enum.count @@
              intersection set @@ Enum.skip 1 @@ Enum.clone l
            in n mod 2 = b)
      |> EnumL.for_all ~f:((=) true))
  |> EnumL.filter ~f:((=) true)
  |> Enum.count
  |> Printf.printf "%d\n"