
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
 $\Phi$  is the union of all triangles that are symmetric of the triangle  $ABC$  wrt a point  $O$ , as point  $O$  moves along the triangle's sides. If the  area of the triangle is  $E$ , find the area of  $\Phi$ .

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