
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
Consider a  $20$ -sided convex polygon  $K$ , with vertices  $A_1, A_2,...,A_{20}$  in that order. Find the number of ways in which three sides of  $K$  can be chosen so that every pair among them has at least two sides of  $K$  between them. (For example  $(A_1A_2, A_4A_5, A_{11}A_{12})$  is an admissible triple while  $(A_1A_2, A_4A_5, A_{19}A_{20})$  is not.

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