
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, points $A$, $B$, $C$, $D$, $E$, and $F$ lie on a straight line with $AB=4$, $BC=3$, $CD=5$, $DE=2$, and $EF=6$. Semicircles with diameters $AF$, $AB$, $BC$, $CD$, $DE$, and $EF$ are drawn. Calculate the area of the shaded region, which is inside the largest semicircle but outside all smaller semicircles.
[asy]
size(6cm); defaultpen(fontsize(10));
pair a = (0, 0); pair b = a + (4, 0); pair c = b + (3, 0); pair d = c + (5, 0); pair e = d + (2, 0); pair f = e + (6, 0);
path region = a{up}..{down}f..{up}e..{down}d..{up}c..{down}b..{up}a--cycle;
filldraw(region, gray(0.75), linewidth(0.75));
draw(a--f, dashed + linewidth(0.75));

// labels
label("$A$", a, S); label("$F$", f, S);
label("$B$", b, N); label("$E$", e, N);
label("$C$", c, S); label("$D$", d, N);
[/asy]

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