
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
In a convex 10-gon \(A_{1} A_{2} \ldots A_{10}\), all sides and all diagonals connecting vertices skipping one (i.e., \(A_{1} A_{3}, A_{2} A_{4},\) etc.) are drawn, except for the side \(A_{1} A_{10}\) and the diagonals \(A_{1} A_{9}\), \(A_{2} A_{10}\).

A path from \(A_{1}\) to \(A_{10}\) is defined as a non-self-intersecting broken line (i.e., a line such that no two nonconsecutive segments share a common point) with endpoints \(A_{1}\) and \(A_{10}\), where each segment coincides with one of the drawn sides or diagonals. Determine the number of such paths.

Present the answer in LaTex format: \boxed{Your answer}
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