
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 quadrilateral  $ABCD$ ,  $AB=2$ ,  $AD=3$ ,  $BC=CD=\sqrt7$ , and  $\angle DAB=60^\circ$ . Semicircles  $\gamma_1$  and  $\gamma_2$  are erected on the exterior of the quadrilateral with diameters  $\overline{AB}$  and  $\overline{AD}$ ; points  $E\neq B$  and  $F\neq D$  are selected on  $\gamma_1$  and  $\gamma_2$  respectively such that  $\triangle CEF$  is equilateral. What is the area of  $\triangle CEF$ ?

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