
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
Consider a layout where a 4-inch by 4-inch square adjoins a 12-inch by 12-inch square. The bottom right corner of the smaller square is attached to the middle of the right edge of the larger square. Determine the area of the shaded region formed where these squares overlap. Refer to the updated drawing below.
[asy]
unitsize(2mm);
defaultpen(linewidth(0.7pt)+fontsize(12pt));

pair H=(0,0), A=(0,12), B=(12,12), C=(12,10), G=(12,0), E=(16,10), F=(16,0);
pair D=extension(A,F,B,G);

draw(A--B--G--H--cycle);
fill(D--C--E--F--cycle, black);
draw(A--F);
draw(G--F--E--C);

label("$A$",A,NW);
label("$H$",H,SW);
label("$B$",B,NE);
label("$C$",C,NE);
label("$G$",G,S);
label("$D$",D,WSW);
label("$E$",E,NE);
label("$F$",F,SE);
[/asy]

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