
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
Consider the following lines: line $l$ is parametrized as:
\[
\begin{align*}
x &= 2 + 3t, \\
y &= 5 - 2t
\end{align*}
\]
and line $m$ is parametrized as:
\[
\begin{align*}
x &= -2 + 3s, \\
y &= 7 - 2s.
\end{align*}
\]
Let $A$ be a point on line $l$, $B$ be a point on line $m$, and let $P$ be the foot of the perpendicular from $A$ to line $m$.

Determine vector $\begin{pmatrix} v_1 \\ v_2 \end{pmatrix}$ such that $\overrightarrow{PA}$ is the projection of $\overrightarrow{BA}$ onto $\begin{pmatrix} v_1 \\ v_2 \end{pmatrix}$ and $3v_1 + 2v_2 = 6$.

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