
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
Let \(P\) be a point on the line
\[
\begin{pmatrix} 1 \\ 3 \\ -2 \end{pmatrix} + u \begin{pmatrix} -1 \\ 3 \\ 2 \end{pmatrix}
\]
and let \(Q\) be a point on the line
\[
\begin{pmatrix} 2 \\ 1 \\ 5 \end{pmatrix} + v \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}.
\]
Find the shortest possible distance \(PQ.\)

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