
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
Felix picks four points uniformly at random inside a unit circle  $\mathcal{C}$ . He then draws the four possible triangles which can be formed using these points as vertices. Finally, he randomly chooses of the six possible pairs of the triangles he just drew.

What is the probability that the center of the circle  $\mathcal{C}$  is contained in the union of the interiors of the two triangles that Felix chose?

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