
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
The matrix for projecting onto a certain line $\ell$ in three-dimensional space, which passes through the origin, is given by
\[
\renewcommand{\arraystretch}{1.5} 
\begin{pmatrix} 
\frac{1}{7} & -\frac{2}{7} & \frac{3}{7} \\
-\frac{2}{7} & \frac{4}{7} & -\frac{6}{7} \\
\frac{3}{7} & -\frac{6}{7} & \frac{9}{7} 
\end{pmatrix} 
\renewcommand{\arraystretch}{1}
.\]
Find the direction vector of line $\ell$. Enter your answer in the form $\begin{pmatrix} a \\ b \\ c \end{pmatrix},$ where $a,$ $b,$ and $c$ are integers, $a > 0,$ and $\gcd(|a|,|b|,|c|) = 1.$

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