
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 that point P is the center of the base A_1 B_1 C_1 of a regular triangular prism A B C-A_1 B_1 C_1, a plane B C D is perpendicular to A P and intersects edge A A_1 at D. If A A_1 = 2 A B = 2, find the volume of the triangular pyramid D-A B C.

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