
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 P and Q are inside triangle ∆ABC, and 
\[
\begin{array}{l}
\overrightarrow{PA} + 2\overrightarrow{PB} + 3\overrightarrow{PC} \\
= 2\overrightarrow{QA} + 3\overrightarrow{QB} + 5\overrightarrow{QC} = \mathbf{0},
\end{array}
\]
find the value of $\frac{|\overrightarrow{PQ}|}{|\overrightarrow{AB}|}$.

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