<p>Your car starts at position <code>0</code> and speed <code>+1</code> on an infinite number line. Your car can go into negative positions. Your car drives automatically according to a sequence of instructions <code>&#39;A&#39;</code> (accelerate) and <code>&#39;R&#39;</code> (reverse):</p>

<ul>
	<li>When you get an instruction <code>&#39;A&#39;</code>, your car does the following:

	<ul>
		<li><code>position += speed</code></li>
		<li><code>speed *= 2</code></li>
	</ul>
	</li>
	<li>When you get an instruction <code>&#39;R&#39;</code>, your car does the following:
	<ul>
		<li>If your speed is positive then <code>speed = -1</code></li>
		<li>otherwise <code>speed = 1</code></li>
	</ul>
	Your position stays the same.</li>
</ul>

<p>For example, after commands <code>&quot;AAR&quot;</code>, your car goes to positions <code>0 --&gt; 1 --&gt; 3 --&gt; 3</code>, and your speed goes to <code>1 --&gt; 2 --&gt; 4 --&gt; -1</code>.</p>

<p>Given a target position <code>target</code>, return <em>the length of the shortest sequence of instructions to get there</em>.</p>

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

<pre>
<strong>Input:</strong> target = 3
<strong>Output:</strong> 2
<strong>Explanation:</strong> 
The shortest instruction sequence is &quot;AA&quot;.
Your position goes from 0 --&gt; 1 --&gt; 3.
</pre>

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

<pre>
<strong>Input:</strong> target = 6
<strong>Output:</strong> 5
<strong>Explanation:</strong> 
The shortest instruction sequence is &quot;AAARA&quot;.
Your position goes from 0 --&gt; 1 --&gt; 3 --&gt; 7 --&gt; 7 --&gt; 6.
</pre>

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

<ul>
	<li><code>1 &lt;= target &lt;= 10<sup>4</sup></code></li>
</ul>

class Solution {
public:
    int racecar(int target) {
        unordered_set<tuple<int, int>, TupleHash> visited;
        queue<tuple<int, int, int>> q;
        q.push({0, 0, 1});
        visited.insert({0, 1});
        while (!q.empty()) {
            auto [step, position, speed] = q.front();
            q.pop();
            if (position == target) {
                return step;
            }
            int nextPos = position + speed;
            int nextSpeed = speed * 2;
            if (abs(nextPos) <= 2 * target && !visited.count({nextPos, nextSpeed})) {
                q.push({step + 1, nextPos, nextSpeed});
                visited.insert({nextPos, nextSpeed});
            }

            // Reverse direction: next state (same position, opposite speed)
            int reverseSpeed = (speed > 0) ? -1 : 1;
            if (!visited.count({position, reverseSpeed})) {
                q.push({step + 1, position, reverseSpeed});
                visited.insert({position, reverseSpeed});
            }
        }
        return -1; // shouldnt be reachable
    }

    struct TupleHash {
        std::size_t operator()(const std::tuple<int, int>& t) const {
            auto hash1 = std::hash<int>{}(std::get<0>(t));
            auto hash2 = std::hash<int>{}(std::get<1>(t));
            return hash1 ^ (hash2 << 1); // Combine the hashes
        }
    };
};

/*



bfs with queue storing: step count, position, speed

*/
150823