
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 $P$ lies within $\triangle ABC$ and satisfies the equation $\overrightarrow{PA} + 3\overrightarrow{PB} + 5\overrightarrow{PC} = \overrightarrow{0}$, let $S_1$, $S_2$, and $S_3$ represent the areas of $\triangle ABP$, $\triangle BCP$, and $\triangle ACP$, respectively. Determine the ratio $S_1 : S_2 : S_3 = \_\_\_\_\_\_\_.$

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