
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
Let  $A$  and  $B$  be points on circle  $\Gamma$  such that  $AB=\sqrt{10}.$  Point  $C$  is outside  $\Gamma$  such that  $\triangle ABC$  is equilateral. Let  $D$  be a point on  $\Gamma$  and suppose the line through  $C$  and  $D$  intersects  $AB$  and  $\Gamma$  again at points  $E$  and  $F \neq D.$  It is given that points  $C, D, E, F$  are collinear in that order and that  $CD=DE=EF.$  What is the area of  $\Gamma?$ *Proposed by Kyle Lee*

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