
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 represents reflection over a line $\ell$ that passes through the origin, given by:
\[\begin{pmatrix} \frac{9}{41} & \frac{40}{41} \\ \frac{40}{41} & -\frac{9}{41} \end{pmatrix}.\]
Find the direction vector of line $\ell$, in the form $\begin{pmatrix} a \\ b \end{pmatrix}$, where $a, b$ are integers, $a > 0$, and $\gcd(|a|, |b|) = 1$.

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