
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
In the triangle  $\triangle ABC$ , let  $G$  be the centroid, and let  $I$  be the center of the inscribed circle.  Let  $\alpha$  and  $\beta$  be the angles at the vertices  $A$  and  $B$ , respectively.  Suppose that the segment  $IG$  is parallel to  $AB$  and that  $\beta = 2\tan^{-1}(1/3)$ .  Find  $\alpha$ .

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