<p>A certain bug&#39;s home is on the x-axis at position <code>x</code>. Help them get there from position <code>0</code>.</p>

<p>The bug jumps according to the following rules:</p>

<ul>
	<li>It can jump exactly <code>a</code> positions <strong>forward</strong> (to the right).</li>
	<li>It can jump exactly <code>b</code> positions <strong>backward</strong> (to the left).</li>
	<li>It cannot jump backward twice in a row.</li>
	<li>It cannot jump to any <code>forbidden</code> positions.</li>
</ul>

<p>The bug may jump forward <strong>beyond</strong> its home, but it <strong>cannot jump</strong> to positions numbered with <strong>negative</strong> integers.</p>

<p>Given an array of integers <code>forbidden</code>, where <code>forbidden[i]</code> means that the bug cannot jump to the position <code>forbidden[i]</code>, and integers <code>a</code>, <code>b</code>, and <code>x</code>, return <em>the minimum number of jumps needed for the bug to reach its home</em>. If there is no possible sequence of jumps that lands the bug on position <code>x</code>, return <code>-1.</code></p>

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

<pre>
<strong>Input:</strong> forbidden = [14,4,18,1,15], a = 3, b = 15, x = 9
<strong>Output:</strong> 3
<strong>Explanation:</strong> 3 jumps forward (0 -&gt; 3 -&gt; 6 -&gt; 9) will get the bug home.
</pre>

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

<pre>
<strong>Input:</strong> forbidden = [8,3,16,6,12,20], a = 15, b = 13, x = 11
<strong>Output:</strong> -1
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> forbidden = [1,6,2,14,5,17,4], a = 16, b = 9, x = 7
<strong>Output:</strong> 2
<strong>Explanation:</strong> One jump forward (0 -&gt; 16) then one jump backward (16 -&gt; 7) will get the bug home.
</pre>

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

<ul>
	<li><code>1 &lt;= forbidden.length &lt;= 1000</code></li>
	<li><code>1 &lt;= a, b, forbidden[i] &lt;= 2000</code></li>
	<li><code>0 &lt;= x &lt;= 2000</code></li>
	<li>All the elements in <code>forbidden</code> are distinct.</li>
	<li>Position <code>x</code> is not forbidden.</li>
</ul>

from collections import deque

class Solution:
    def minimumJumps(self, forbidden: list[int], a: int, b: int, x: int) -> int:
        # Set up forbidden set for quick lookup
        forbidden_set = set(forbidden)
        max_position = 6000  # Maximum range to explore based on constraints

        # BFS queue: (current_position, number_of_jumps, last_jump_was_backward)
        queue = deque([(0, 0, False)])
        
        # Visited set to track visited positions
        visited = set([(0, False)])

        while queue:
            position, jumps, last_backward = queue.popleft()

            # If we reached the home position
            if position == x:
                return jumps

            # Try forward jump
            forward_position = position + a
            if forward_position <= max_position and forward_position not in forbidden_set and (forward_position, False) not in visited:
                visited.add((forward_position, False))
                queue.append((forward_position, jumps + 1, False))

            # Try backward jump if the last jump wasn't backward
            if not last_backward:
                backward_position = position - b
                if backward_position >= 0 and backward_position not in forbidden_set and (backward_position, True) not in visited:
                    visited.add((backward_position, True))
                    queue.append((backward_position, jumps + 1, True))

        # If we exhaust the queue and never reach the home position
        return -1

# Example usage:
sol = Solution()
print(sol.minimumJumps([14, 4, 18, 1, 15], 3, 15, 9))  # Output: 3

18400