
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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