
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 \( P \) is inside \( \triangle ABC \), and satisfies
$$
\overrightarrow{AP}=\frac{1}{3} \overrightarrow{AB}+\frac{1}{4} \overrightarrow{AC} ,
$$
and let the areas of \( \triangle PBC \), \( \triangle PCA \), and \( \triangle PAB \) be \( S_{1}, S_{2}, S_{3} \) respectively. Determine the ratio \( S_{1}: S_{2}: S_{3} \).

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