
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  $A$ ,  $B$  be opposite vertices of a unit square with circumcircle  $\Gamma$ . Let  $C$  be a variable point on  $\Gamma$ . If  $C\not\in\{A, B\}$ , then let  $\omega$  be the incircle of triangle  $ABC$ , and let  $I$  be the center of  $\omega$ . Let  $C_1$  be the point at which  $\omega$  meets  $\overline{AB}$ , and let  $D$  be the reflection of  $C_1$  over line  $CI$ . If  $C \in\{A, B\}$ , let  $D = C$ . As  $C$  varies on  $\Gamma$ ,  $D$  traces out a curve  $\mathfrak C$  enclosing a region of area  $\mathcal A$ . Compute  $\lfloor 10^4 \mathcal A\rfloor$ . 

*Proposed by Brandon Wang*

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