
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 point \( O \) is inside triangle \( \triangle ABC \), and \( 3 \overrightarrow{AB} + 2 \overrightarrow{BC} + \overrightarrow{CA} = 4 \overrightarrow{AO} \), let the area of \( \triangle ABC \) be \( S_1 \) and the area of \( \triangle OBC \) be \( S_2 \). Find the value of \( \frac{S_1}{S_2} \).

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