
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 
\[
\begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} + t \begin{pmatrix} 3 \\ -3 \\ 2 \end{pmatrix}
\]
and let $Q$ be a point on the line
\[
\begin{pmatrix} -1 \\ -1 \\ 5 \end{pmatrix} + s \begin{pmatrix} 2 \\ 4 \\ -3 \end{pmatrix}.
\]
Additionally, suppose $z$-coordinate of point $P$ equals $0$. Find the shortest possible distance $PQ$.

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