<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<string> visited; // Set to track visited states
        queue<pair<int, int>> q; // Queue for BFS

        q.push({0, 1}); // Start from position 0 with speed 1
        visited.insert("0_1"); // Mark the initial state as visited

        int moves = 0;

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

            for (int i = 0; i < n; ++i) {
                auto [position, speed] = q.front();
                q.pop();

                // Move forward
                int new_position = position + speed;
                int new_speed = speed * 2;

                // Check speed constraints
                if (new_speed < 10000 && new_speed > -10000) {
                    string forward_state = to_string(new_position) + "_" + to_string(new_speed);
                    
                    if (new_position == target) return moves;
                    if (!visited.count(forward_state)) {
                        q.push({new_position, new_speed});
                        visited.insert(forward_state);
                    }
                }

                // Reverse direction
                new_position = position;
                new_speed = speed > 0 ? -1 : 1;
                string reverse_state = to_string(new_position) + "_" + to_string(new_speed);

                if (!visited.count(reverse_state)) {
                    q.push({new_position, new_speed});
                    visited.insert(reverse_state);
                }
            }
        }

        return 0;
    }
};

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