
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 \( P \) be a point inside triangle \( ABC \) such that
\[ 2\overrightarrow{PA} + 3\overrightarrow{PB} + \overrightarrow{PC} = \mathbf{0}. \]
Find the ratio of the area of triangle \( ABC \) to the area of triangle \( APB \).

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