<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:

    struct hash_pair { 
        template <class T1, class T2> 
        size_t operator()(const pair<T1, T2>& p) const { 
            auto hash1 = hash<T1>{}(p.first); 
            auto hash2 = hash<T2>{}(p.second); 
            return hash1 ^ hash2; 
        } 
    };

    int racecar(int target) {
        target = abs(target); // We can focus on positive target due to symmetry
        queue< pair<int, int> > q; // Stores (position, speed)
        unordered_set< pair<int, int>, hash_pair> visited; // Keeps track of visited states

        q.push({0, 1}); // Start at position 0 with speed 1
        visited.insert({0, 1});
        
        int steps = 0;

        while (!q.empty()) {
            int size = q.size();

            while (size--) {
                auto [pos, speed] = q.front();
                q.pop();

                if (pos == target) {
                    return steps;
                }

                // Accelerate
                long long newPos = (long long)pos + (long long)speed + 0LL;
                long long newSpeed = speed * 2LL;
                if (!visited.count({newPos, newSpeed}) and !(newPos<0 and newSpeed<0) and newSpeed < INT_MAX and newPos < INT_MAX) {
                    q.push({newPos, newSpeed});
                    visited.insert({newPos, newSpeed});
                }

                // Reverse
                newPos = pos;
                newSpeed = speed > 0 ? -1 : 1;
                if (!visited.count({newPos, newSpeed}) and !(newPos<0 and newSpeed<0) and newSpeed < INT_MAX and newPos < INT_MAX ) {
                    q.push({newPos, newSpeed});
                    visited.insert({newPos, newSpeed});
                }
            }
            steps++;
        }

        return -1; // If the target is unreachable, which shouldn't happen in this problem
    }
};
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