
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
A matrix projects onto a line $\ell$, passing through the origin, is given by
\[
\renewcommand{\arraystretch}{1.5}
\begin{pmatrix}
\frac{1}{18} & -\frac{1}{9} & \frac{1}{6} \\
-\frac{1}{9} & \frac{4}{18} & -\frac{1}{3} \\
\frac{1}{6} & -\frac{1}{3} & \frac{1}{2}
\end{pmatrix}
\renewcommand{\arraystretch}{1}
.\]
Find the direction vector of line $\ell$. Enter your answer as $\begin{pmatrix} a \\ b \\ c \end{pmatrix}$, where $a$, $b$, $c$ are integers, $a > 0$, and $\gcd(|a|,|b|,|c|) = 1$.

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