
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 10 inches is shown. If $P$ is a point such that the segments $\overline{PA}$, $\overline{PB}$, and $\overline{PC}$ are equal in length, and segment $\overline{PC}$ is perpendicular to segment $\overline{GD}$, what is the area, in square inches, of triangle $APB$? Here, $G$ is the midpoint of side $AD$.
[asy]
pair A, B, C, D, G, P;
A = (0,0); B= (10,0); C = (5,10); D = (10,10); G = (0,10); P = (5,5);
draw(A--B--D--G--cycle);
draw(C--P); draw(P--A); draw(P--B);
label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$D$",D,NE); label("$G$",G,NW);label("$P$",P,NW);
label("$10''$",(5,0),S);
[/asy]

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