
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
The center of circle $k$ is $O$. Points $A, B, C, D$ are consecutive points on the circumference of $k$ such that $\angle AOB = \angle BOC = \angle COD = \alpha < 60^\circ$. Point $E$ is the projection of $D$ onto the diameter $AO$. Point $F$ is the trisection point of segment $DE$ closer to $E$. Finally, $G$ is the intersection point of $AO$ and $BF$. What is the measure of angle $OGD$ as $\alpha \rightarrow 60^\circ$?

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