
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
Circle  $\Gamma$  has diameter  $\overline{AB}$  with  $AB = 6$ .  Point  $C$  is constructed on line  $AB$  so that  $AB = BC$  and  $A \neq C$ . Let  $D$  be on  $\Gamma$  so that  $\overleftrightarrow{CD}$  is tangent to  $\Gamma$ . Compute the distance from line  $\overleftrightarrow{AD}$  to the circumcenter of  $\triangle ADC$ .

*Proposed by Justin Hsieh*

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