
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 \(\mathcal{P}\) be a regular 17-gon. We draw in the \(\binom{17}{2}\) diagonals and sides of \(\mathcal{P}\) and paint each side or diagonal one of eight different colors. Suppose that there is no triangle (with vertices among vertices of \(\mathcal{P}\)) whose three edges all have the same color. What is the maximum possible number of triangles, all of whose edges have different colors?

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