
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 triangular prism \( ABC-A_1B_1C_1 \) with a height of 2 and a base edge length of 1, the centroid of the top triangular face \( \triangle A_1B_1C_1 \) is denoted as \( P \). A plane \( BCD \) is drawn perpendicular to \( AP \) and intersects the edge \( AA_1 \) at point \( D \). Find the area of the cross-section \( \triangle BCD \).

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