
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 $\ell,$ which passes through the origin, is given by
\[
\renewcommand{\arraystretch}{1.5} 
\begin{pmatrix} 
\frac{1}{6} & -\frac{1}{18} & -\frac{1}{2} \\ 
-\frac{1}{18} & \frac{1}{72} & \frac{1}{12} \\ 
-\frac{1}{2} & \frac{1}{12} & \frac{11}{12} 
\end{pmatrix} 
\renewcommand{\arraystretch}{1}
.\]
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}
user