
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  $O$  and  $A$  be two points in the plane with  $OA = 30$ , and let  $\Gamma$  be a circle with center  $O$  and radius  $r$ . Suppose that there exist two points  $B$  and  $C$  on  $\Gamma$  with  $\angle ABC = 90^{\circ}$  and  $AB = BC$ . Compute the minimum possible value of  $\lfloor r \rfloor.$ 

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