
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 $P$ inside $\triangle ABC$, such that
$$
\overrightarrow{AP} = \frac{1}{3} \overrightarrow{AB} + \frac{1}{4} \overrightarrow{AC},
$$

and let the areas of $\triangle PBC$, $\triangle PCA$, and $\triangle PAB$ be $S_{1}$, $S_{2}$, and $S_{3}$ respectively. Then determine the ratio $S_{1} : S_{2} : S_{3}$.

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