
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 the point \( P \) inside the triangle \( \triangle ABC \), satisfying \( \overrightarrow{AP} = \frac{1}{3} \overrightarrow{AB} + \frac{1}{4} \overrightarrow{AC} \), let the areas of triangles \( \triangle PBC \), \( \triangle PCA \), and \( \triangle PAB \) be \( S_1 \), \( S_2 \), and \( S_3 \) respectively. Determine the ratio \( S_1 : S_2 : S_3 = \quad \).

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