
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 $l$ parametrized as
\begin{align*} 
x &= 2 + 2t,\\
y &=  3 + t
\end{align*}
and the line $m$ parametrized as
\begin{align*} 
x &= -3 + 2s,\\
y &=  5 + s.
\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 $v_1+v_2 = 3$. Find $\begin{pmatrix}v_1 \\ v_2 \end{pmatrix}$.

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