
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 10 inches is shown. If \( P \) is a point such that the segments \( \overline{PA} \), \( \overline{PB} \), and \( \overline{PC} \) are equal in length, where \( A \) and \( B \) are the midpoints of adjacent sides of the square, and \( \overline{PC} \) is perpendicular to a side of the square, find the area, in square inches, of triangle \( APB \).

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