
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
Let \( P \) be a set of \( n \geq 3 \) points in the plane, no three of which are on a line. How many possibilities are there to choose a set \( T \) of \( \binom{n-1}{2} \) triangles, whose vertices are all in \( P \), such that each triangle in \( T \) has a side that is not a side of any other triangle in \( T \)?

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