
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
Quadrilateral  $ABCD$  is inscribed in circle  $\Gamma$ . Segments  $AC$  and  $BD$  intersect at  $E$ . Circle  $\gamma$  passes through  $E$  and is tangent to  $\Gamma$  at  $A$ . Suppose the circumcircle of triangle  $BCE$  is tangent to  $\gamma$  at  $E$  and is tangent to line  $CD$  at  $C$ . Suppose that  $\Gamma$  has radius  $3$  and  $\gamma$  has radius  $2$ . Compute  $BD$ .

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