
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 two 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 &= -4 + 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 $m$.

Then $\overrightarrow{PA}$ is the projection of $\overrightarrow{BA}$ onto some vector $\begin{pmatrix} v_1\\v_2\end{pmatrix}$ such that $v_1 + v_2 = 1$. Find $\begin{pmatrix} v_1 \\ v_2 \end{pmatrix}$.

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