
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
\[\renewcommand{\arraystretch}{1.5}\begin{pmatrix} \frac{1}{5} & -\frac{1}{10} & \frac{1}{10} \\ -\frac{1}{10} & \frac{2}{15} & -\frac{1}{15} \\ \frac{1}{10} & -\frac{1}{15} & \frac{4}{15} \end{pmatrix}\renewcommand{\arraystretch}{1}.\]
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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