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 JavaScript (node.js v5.12) (judge verdict: Accepted)

"use strict";

function _toArray(arr) { return _arrayWithHoles(arr) || _iterableToArray(arr) || _nonIterableRest(); }

function _iterableToArray(iter) { if (Symbol.iterator in Object(iter) || Object.prototype.toString.call(iter) === "[object Arguments]") return Array.from(iter); }

function _slicedToArray(arr, i) { return _arrayWithHoles(arr) || _iterableToArrayLimit(arr, i) || _nonIterableRest(); }

function _nonIterableRest() { throw new TypeError("Invalid attempt to destructure non-iterable instance"); }

function _iterableToArrayLimit(arr, i) { var _arr = []; var _n = true; var _d = false; var _e = undefined; try { for (var _i = arr[Symbol.iterator](), _s; !(_n = (_s = _i.next()).done); _n = true) { _arr.push(_s.value); if (i && _arr.length === i) break; } } catch (err) { _d = true; _e = err; } finally { try { if (!_n && _i["return"] != null) _i["return"](); } finally { if (_d) throw _e; } } return _arr; }

function _arrayWithHoles(arr) { if (Array.isArray(arr)) return arr; }

!function () {
  function splitNum(line) {
    return (Array.isArray(line) ? line : line.split(/\s+/)).map(function (n) {
      return parseInt(n);
    });
  }

  !function (input) {
    var lines = input.split("\n");

    var _splitNum = splitNum(lines[0]),
        _splitNum2 = _slicedToArray(_splitNum, 2),
        N = _splitNum2[0],
        M = _splitNum2[1];

    var lights = lines.slice(1, M + 1).map(function (line, i) {
      var _splitNum3 = splitNum(line),
          _splitNum4 = _toArray(_splitNum3),
          k = _splitNum4[0],
          s = _splitNum4.slice(1);

      return {
        k: k,
        s: s,
        li: i
      };
    });
    var ps = splitNum(lines[M + 1]);
    var count = 0;

    ll: for (var i = 0; i < Math.pow(2, N); i++) {
      var _iteratorNormalCompletion = true;
      var _didIteratorError = false;
      var _iteratorError = undefined;

      try {
        for (var _iterator = lights[Symbol.iterator](), _step; !(_iteratorNormalCompletion = (_step = _iterator.next()).done); _iteratorNormalCompletion = true) {
          var _step$value = _step.value,
              s = _step$value.s,
              li = _step$value.li;
          var swc = 0;
          var _iteratorNormalCompletion2 = true;
          var _didIteratorError2 = false;
          var _iteratorError2 = undefined;

          try {
            for (var _iterator2 = s[Symbol.iterator](), _step2; !(_iteratorNormalCompletion2 = (_step2 = _iterator2.next()).done); _iteratorNormalCompletion2 = true) {
              var si = _step2.value;
              if (i & 1 << si - 1) swc++;
            }
          } catch (err) {
            _didIteratorError2 = true;
            _iteratorError2 = err;
          } finally {
            try {
              if (!_iteratorNormalCompletion2 && _iterator2.return != null) {
                _iterator2.return();
              }
            } finally {
              if (_didIteratorError2) {
                throw _iteratorError2;
              }
            }
          }

          if (swc % 2 !== ps[li]) continue ll;
        }
      } catch (err) {
        _didIteratorError = true;
        _iteratorError = err;
      } finally {
        try {
          if (!_iteratorNormalCompletion && _iterator.return != null) {
            _iterator.return();
          }
        } finally {
          if (_didIteratorError) {
            throw _iteratorError;
          }
        }
      }

      count++;
    }

    console.log(count);
  }(require("fs").readFileSync("/dev/stdin", "utf8"));
}();