<p>An <strong>n-bit gray code sequence</strong> is a sequence of <code>2<sup>n</sup></code> integers where:</p>

<ul>
	<li>Every integer is in the <strong>inclusive</strong> range <code>[0, 2<sup>n</sup> - 1]</code>,</li>
	<li>The first integer is <code>0</code>,</li>
	<li>An integer appears <strong>no more than once</strong> in the sequence,</li>
	<li>The binary representation of every pair of <strong>adjacent</strong> integers differs by <strong>exactly one bit</strong>, and</li>
	<li>The binary representation of the <strong>first</strong> and <strong>last</strong> integers differs by <strong>exactly one bit</strong>.</li>
</ul>

<p>Given an integer <code>n</code>, return <em>any valid <strong>n-bit gray code sequence</strong></em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> n = 2
<strong>Output:</strong> [0,1,3,2]
<strong>Explanation:</strong>
The binary representation of [0,1,3,2] is [00,01,11,10].
- 0<u>0</u> and 0<u>1</u> differ by one bit
- <u>0</u>1 and <u>1</u>1 differ by one bit
- 1<u>1</u> and 1<u>0</u> differ by one bit
- <u>1</u>0 and <u>0</u>0 differ by one bit
[0,2,3,1] is also a valid gray code sequence, whose binary representation is [00,10,11,01].
- <u>0</u>0 and <u>1</u>0 differ by one bit
- 1<u>0</u> and 1<u>1</u> differ by one bit
- <u>1</u>1 and <u>0</u>1 differ by one bit
- 0<u>1</u> and 0<u>0</u> differ by one bit
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> n = 1
<strong>Output:</strong> [0,1]
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= 16</code></li>
</ul>

class Solution {
public:

    //1: normal node
    //2: end node
    //-1: passed normal node
    //-2: passed end node
    //0: start node
    bool dfs(vector<int>& path, int states[], int n, int num){
        // for(int p: path){
        //     printf("%d, ", p);
        // }
        // printf("\n");
        if(path.size() == pow(2, n) && states[num] == -2){
            return true;
        }
        int mask = 1;
        for(int i=0;i<n;i++){
            int next = num ^ mask;
            // printf("check %d, %d, %d\n", num, mask, next);
            if(states[next] <= 0){
                mask <<= 1;
                continue;
            }
            states[next] *= -1;
            path.push_back(next);
            if(dfs(path, states, n, next)){
                return true;
            }
            path.pop_back();
            states[next] *= -1;
            mask <<= 1;
        }
        return false;
    }


    vector<int> grayCode(int n) {
        int states[int(pow(2, n))];
        vector<int> path = {0};
        int mask = 1;
        for(int i=0;i<pow(2, n);i++){
            states[i] = 1;
        }
        for(int i=0;i<n;i++){
            states[mask] = 2;
            mask <<= 1;
        }
        states[0] = 0;

        dfs(path, states, n, 0);
        return path;
        
    }


};
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