
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
What is the area of the hexagon depicted below, which is constructed with several right-angled triangles and a central square?

[asy]
unitsize(1cm);
defaultpen(linewidth(0.7)+fontsize(10));
dotfactor = 4;

real w=1, h=1;
pair A=(0,0), B=(2*w, 2*h), C=(B.x, 0), D=(2*w+h, 2*w), E=(w+h, -2*h), F=(-w, -h);

// Hexagon Points
pair G=(0,2*h), H=(w, 2*h), I=(2*w, h), J=(w+h, -h);

draw(A--G--H--B--I--C--J--E--F--cycle);
fill(A--G--H--B--I--C--J--E--F--cycle, lightgray);

// Square in the center
draw(G--H--I--J--cycle, blue+linewidth(1));
fill(G--H--I--J--blue, opacity=0.5);

label("A", A, SW);
label("B", B, NE);
label("C", C, SE);
label("D", D, NE);
label("E", E, SE);
label("F", F, SW);
label("G", G, NW);
label("H", H, NE);
label("I", I, E);
label("J", J, S);

draw((A--B--C--A));
draw((D--E--F--D));

label("2 units", (A+B)/2, N);
label("1 unit", (B+C)/2, E);
label("1 unit", (C+E)/2, S);
label("1 unit", (E+A)/2, W);

[/asy]

The side lengths of the square in the center are all 1 unit. Distances along the outer portions (from hexagon vertice to vertice, clockwise) are as follows: from point A to B (along the hexagon) is 2 units, B to C is 1 unit, then similarly for other sides by symmetry.

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