
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
A square with sides of length 8 inches is given. If \( P \) is a point within the square such that segments \(\overline{PA}\), \(\overline{PB}\), and \(\overline{PC}\) are equal in length and segment \(\overline{PD}\) is perpendicular to segment \(\overline{AB}\), find the area, in square inches, of triangle \( APB \). Assume point \( A \) is the bottom-left corner, \( B \) is the bottom-right corner, \( C \) is the top-right corner, and \( D \) is the top-left corner of the square.

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