<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:

   int binpow(int a, int b){
        if(b == 0) return 1;
        int v = binpow(a, b / 2);
        if(b & 1) return a * v * v;
        else return v * v;
   }

   // Variables to track state
   vector<int> numbers;
   vector<int> sol, ans;
   bool done = false;
   int N;
    
    // Function to check if val differs from the last value by exactly one bit
    // bool differsByOneBit(int val1, int val2){
    //     int xorValue = val1 ^ val2;
    //     return (xorValue & (xorValue - 1)) == 0; // Checks if xorValue is a power of 2
    // }

    // Recursive function to generate Gray code
    void rec(int level, int n){
        if(done) return;

        if(level == N){
            done = true;
            sol = ans;
            return;
        }
       
        for(int i = 0; i < n; i++){
            int val = ans.back() ^ (1 << i); // Toggle the ith bit
            if(!numbers[val]){
                // Place the value
                ans.push_back(val);
                numbers[val] = 1;
                // Move to the next level
                rec(level + 1, n);
                // Unplace the value (backtrack)
                ans.pop_back();
                numbers[val] = 0;
            }
        }
    }

    vector<int> grayCode(int n) {
        N = binpow(2, n);
        numbers.assign(N, 0);
        // First integer is "0"
        ans.push_back(0);
        numbers[0] = 1;

        rec(1, n); // Start recursion from level 1
        return sol;
    }
};

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