
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
**(a)** What is the maximal number of acute angles in a convex polygon?**(b)** Consider  $m$  points in the interior of a convex  $n$ -gon. The  $n$ -gon is partitioned into triangles whose vertices are among the  $n + m$  given points (the vertices of the  $n$ -gon and the given points). Each of the  $m$  points in the interior is a vertex of at least one triangle. Find the number of triangles obtained.

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