
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
Given that \( O \) is the circumcenter of the acute-angled \( \triangle ABC \), and the areas of \( \triangle BOC \), \( \triangle COA \), and \( \triangle AOB \) form an arithmetic sequence, find the minimum value of \( \tan A + 2 \tan C \).

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