
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
Triangle  $\triangle ABC$  has circumcenter  $O$  and incircle  $\gamma$ . Suppose that  $\angle BAC =60^\circ$  and  $O$  lies on  $\gamma$ . If \[ \tan B \tan C = a + \sqrt{b} \] for positive integers  $a$  and  $b$ , compute  $100a+b$ .

*Proposed by Kaan Dokmeci*

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