
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
Given a triangular pyramid \( P-ABC \) with each side of the base measuring 1 and height \(\sqrt{2}\), and its inscribed sphere has a center \( O \) and radius \( r \). Find the area of the region in the base \( ABC \) such that every point in this region is at a distance not greater than \( 2r \) from point \( O \).

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