
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 $EFGH$ is inscribed in the region bound by the parabola $y = x^2 - 8x + 16$ and the $x$-axis. Find the area of square $EFGH$.

[asy]
unitsize(0.8 cm);

real parab2 (real x) {
  return(x^2 - 8*x + 16);
}

pair E, F, G, H;
real x = -2 + 2*sqrt(3);

E = (4 - x,0);
F = (4 + x,0);
G = (4 + x,-2*x);
H = (4 - x,-2*x);

draw(graph(parab2,0.5,7.5));
draw(E--H--G--F);
draw((0,0)--(8,0));

label("$E$", E, N);
label("$F$", F, N);
label("$G$", G, SE);
label("$H$", H, SW);
[/asy]

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