
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 a point \( O \) inside triangle \( \triangle ABC \), where \( \overrightarrow{OA} + 2\overrightarrow{OB} + 3\overrightarrow{OC} = \mathbf{0} \), determine the ratio of the area of \( \triangle ABC \) to the area of \( \triangle AOC \).

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