
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 $Q$ is a point such that the segments $\overline{QA}$, $\overline{QB}$, $\overline{QC}$ are equal in length, and segment $\overline{QC}$ is perpendicular to segment $\overline{HD}$, find the area, in square inches, of triangle $AQB$. [asy]
pair A, B, C, D, H, Q;
A = (0,0); B= (2,0); C = (1,2); D = (2,2); H = (0,2); Q = (1,1);
draw(A--B--D--H--cycle);
draw(C--Q); draw(Q--A); draw(Q--B);
label("$A$",A,SW); label("$B$",B,SE);label("$C$",C,N);label("$D$",D,NE);label("$Q$",Q,NW);label("$H$",H,NW);
label("$8''$",(1,0),S);

[/asy]

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