
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 bounded by the parabola $y = x^2 - 6x + 5$ and the $x$-axis. Find the area of square $EFGH$.
\[
\text{[asy]}
unitsize(0.8 cm);

real parab (real x) {
  return(x^2 - 6*x + 5);
}

pair E, F, G, H;
real x = -1 + sqrt(3);

E = (3 - x,0);
F = (3 + x,0);
G = (3 + x,-2*x);
H = (3 - x,-2*x);

draw(graph(parab,0,6));
draw(E--H--G--F);
draw((0,0)--(6,0));

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

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