
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 such that the segments $\overline{PA}$, $\overline{PB}$, $\overline{PC}$, and $\overline{PD}$ are equal in length, and segment $\overline{PC}$ is perpendicular to $\overline{FD}$, what is the area, in square inches, of quadrilateral $APBD$?
[asy]
pair A, B, C, D, F, P;
A = (0,0); B= (2,0); C = (1,2); D = (2,2); F = (0,2); P = (1,1);
draw(A--B--D--F--cycle);
draw(C--P); draw(P--A); draw(P--B); draw(P--D);
label("$A$",A,SW); label("$B$",B,SE);label("$C$",C,N);label("$D$",D,NE);label("$F$",F,NW);label("$P$",P,center);
label("$8''$",(1,0),S);
[/asy]

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