
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 point \( O \) be a point inside triangle \( ABC \) that satisfies the equation
\[
\overrightarrow{OA} + 2 \overrightarrow{OB} + 3 \overrightarrow{OC} = 3 \overrightarrow{AB} + 2 \overrightarrow{BC} + \overrightarrow{CA}.
\]

Then, find the value of \(\frac{S_{\triangle AOB} + 2 S_{\triangle BOC} + 3 S_{\triangle COA}}{S_{\triangle ABC}}\).

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