
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
In the diagram below, a 4-inch by 4-inch square adjoins a 12-inch by 12-inch square. What is the area of the shaded region? Express your answer in square inches as a common fraction.
[asy]
unitsize(2mm);
defaultpen(linewidth(0.7pt)+fontsize(12pt));

pair H=(0,0), A=(0,12), B=(12,12), C=(12,4), G=(12,0), E=(16,4), 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,E);
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