
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
Eight points are chosen on the circumference of a circle, labeled \( P_{1}, P_{2}, \ldots, P_{8} \) in clockwise order. A route is a sequence of at least two points \( P_{a_{1}}, P_{a_{2}}, \ldots, P_{a_{n}} \) such that if an ant were to visit these points in their given order, starting at \( P_{a_{1}} \) and ending at \( P_{a_{n}} \), by following \( n-1 \) straight line segments (each connecting \( P_{a_{i}} \) and \( P_{a_{i+1}} \) ), it would never visit a point twice or cross its own path. Find the number of routes.

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