
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  $G$  be the centroid of triangle  $ABC$ . Find the biggest  $\alpha$  such that there exists a triangle for which there are at least three angles among  $\angle GAB, \angle GAC, \angle GBA, \angle GBC, \angle GCA, \angle GCB$  which are  $\geq \alpha$ .

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