
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 8 inches is shown. If $P$ is a point within the square such that the segments $\overline{PA}$, $\overline{PB}$, and $\overline{PD}$ are equal in length, and segment $\overline{PD}$ is perpendicular to segment $\overline{BC}$, find the area, in square inches, of triangle $\triangle APB$. [asy]
pair A, B, C, D, P;
A = (0,0); B= (8,0); C = (8,8); D = (0,8); P = (4,4);
draw(A--B--C--D--cycle);
draw(P--D); draw(P--A); draw(P--B);
label("$A$",A,SW); label("$B$",B,SE);label("$C$",C,NE);label("$D$",D,NW);label("$P$",P,N);
label("$8''$",(4,0),S);
[/asy]

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