
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
Consider a square with sides of 10 inches. Let $P$ be a point inside the square such that the distances from $P$ to three vertices $A$, $B$, and $D$ of the square are equal, and the segment $\overline{PC}$ is perpendicular to $\overline{AB}$, and $\overline{AB}$ is the bottom side of the square. Determine the area, in square inches, of triangle $APB$.

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