
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  $AXYZB$  be a regular pentagon with area  $5$  inscribed in a circle with center  $O$ .  Let  $Y'$  denote the reflection of  $Y$  over  $\overline{AB}$  and suppose  $C$  is the center of a circle passing through  $A$ ,  $Y'$  and  $B$ . Compute the area of triangle  $ABC$ .

*Proposed by Evan Chen*

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