
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
Point \( O \) is inside triangle \( \triangle ABC \) such that \( \overrightarrow{OA} + 2 \overrightarrow{OB} + 3 \overrightarrow{OC} = \mathbf{0} \). Find the ratio of the area of \( \triangle ABC \) to the area of \( \triangle AOC \).

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