
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  $A_1A_2...A_{101}$  be a regular  $101$ -gon, and colour every vertex red or blue. Let  $N$  be the number of obtuse triangles satisfying the following: The three vertices of the triangle must be vertices of the  $101$ -gon, both the vertices with acute angles have the same colour, and the vertex with obtuse angle have different colour. $(1)$  Find the largest possible value of  $N$ . $(2)$  Find the number of ways to colour the vertices such that maximum  $N$  is acheived. (Two colourings a different if for some  $A_i$  the colours are different on the two colouring schemes).

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