
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 regular tetrahedron \( P-ABC \) with a base edge length of 1 and a height of \( \sqrt{2} \), find the area of the shape obtained by intersecting the sphere centered at the incenter of the tetrahedron with radius \( 2r \) with the base triangle \( \triangle ABC \).

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