
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
\[
\begin{align*} 
x &= 2 + 5t,\\
y &=  3 + 4t,
\end{align*}
\]
and line \( m' \) parametrized as
\[
\begin{align*} 
x &=-7  + 5s,\\
y &=  8 + 4s.
\end{align*}
\]
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 \( \overrightarrow{P'A'} \) is the projection of \( \overrightarrow{B'A'} \) onto some vector \( \begin{pmatrix} v_1\\v_2\end{pmatrix} \) such that \( v_1 + v_2 = 7 \). Find \( \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} \).

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