
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 points $\mathrm{P}$ and $\mathbf{Q}$ are inside $\triangle \mathbf{A B C}$ and $\overrightarrow{P A} + 2 \overrightarrow{P B} + 3 \overrightarrow{P C} = 2 \overrightarrow{Q A} + 3 \overrightarrow{P B} + 5 \overrightarrow{P C} = \overrightarrow{0}$, calculate the value of $\frac{|\overrightarrow{P Q}|}{|\overrightarrow{A B}|}$.

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