
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 $l_1$ parametrized as
\begin{align*} 
x &= 2 + 5t,\\
y &=  3 + 4t
\end{align*}
and line $m_1$ parametrized as
\begin{align*} 
x &= -2 + 5s,\\
y &=  5 + 4s.
\end{align*}
Let $C$ be a point on line $l_1$, $D$ be a point on line $m_1$, and let $Q$ be the foot of the perpendicular from $C$ to line $m_1$.

Then $\overrightarrow{QC}$ is the projection of $\overrightarrow{DC}$ onto some vector $\begin{pmatrix} w_1\\w_2\end{pmatrix}$ such that $w_1+w_2 = 3$. Find $\begin{pmatrix}w_1 \\ w_2 \end{pmatrix}$.

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