
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
Find the direction vector of line $\ell$ which passes through the origin, given that the matrix for projecting onto line $\ell$ is \[\begin{pmatrix} \frac{3}{13} & \frac{2}{13} & \frac{6}{13} \\ \frac{2}{13} & \frac{12}{13} & -\frac{5}{13} \\ \frac{6}{13} & -\frac{5}{13} & \frac{1}{13} \end{pmatrix}.\] Write your answer as $\begin{pmatrix} a \\ b \\ c \end{pmatrix}$, where $a$, $b$, and $c$ are integers and $\gcd(|a|,|b|,|c|) = 1$.

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