
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 triangle $DEF$ has its perpendicular bisectors intersecting its circumcircle at points $D'$, $E''$ (just slightly off the midpoint of arc $DF$), and $F'$, as shown below. The triangle's perimeter is 30 and the radius of the circumcircle is 6. Determine the area of hexagon $DE'F'D''EF'$ given these specific conditions.

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