
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 plane has a special point  $O$  called the origin. Let  $P$  be a set of 2021 points in the plane such that


- no three points in  $P$  lie on a line and
- no two points in  $P$  lie on a line through the origin.

A triangle with vertices in  $P$  is *fat* if  $O$  is strictly inside the triangle. Find the maximum number of fat triangles.

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