<p>There is a regular convex polygon with <code>n</code> vertices. The vertices are labeled from <code>0</code> to <code>n - 1</code> in a clockwise direction, and each vertex has <strong>exactly one monkey</strong>. The following figure shows a convex polygon of <code>6</code> vertices.</p> <img alt="" src="https://assets.leetcode.com/uploads/2023/01/22/hexagon.jpg" style="width: 300px; height: 293px;" /> <p>Each monkey moves simultaneously to a neighboring vertex. A neighboring vertex for a vertex <code>i</code> can be:</p> <ul> <li>the vertex <code>(i + 1) % n</code> in the clockwise direction, or</li> <li>the vertex <code>(i - 1 + n) % n</code> in the counter-clockwise direction.</li> </ul> <p>A <strong>collision</strong> happens if at least two monkeys reside on the same vertex after the movement or intersect on an edge.</p> <p>Return <em>the number of ways the monkeys can move so that at least <strong>one collision</strong></em> <em> happens</em>. Since the answer may be very large, return it modulo <code>10<sup>9 </sup>+ 7</code>.</p> <p><strong>Note</strong> that each monkey can only move once.</p> <p> </p> <p><strong class="example">Example 1:</strong></p> <pre> <strong>Input:</strong> n = 3 <strong>Output:</strong> 6 <strong>Explanation:</strong> There are 8 total possible movements. Two ways such that they collide at some point are: - Monkey 1 moves in a clockwise direction; monkey 2 moves in an anticlockwise direction; monkey 3 moves in a clockwise direction. Monkeys 1 and 2 collide. - Monkey 1 moves in an anticlockwise direction; monkey 2 moves in an anticlockwise direction; monkey 3 moves in a clockwise direction. Monkeys 1 and 3 collide. It can be shown 6 total movements result in a collision. </pre> <p><strong class="example">Example 2:</strong></p> <pre> <strong>Input:</strong> n = 4 <strong>Output:</strong> 14 <strong>Explanation:</strong> It can be shown that there are 14 ways for the monkeys to collide. </pre> <p> </p> <p><strong>Constraints:</strong></p> <ul> <li><code>3 <= n <= 10<sup>9</sup></code></li> </ul>