
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 \( O \) be the circumcenter of an acute triangle \( \triangle ABC \). The areas of \( \triangle BOC \), \( \triangle COA \), and \( \triangle AOB \) form an arithmetic sequence. Find the minimum value of \( \tan A + 3 \tan C \) and determine the measures of the three interior angles at this point.

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