
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 point \( M \) in space is given by the position vector \(\vec{a} = \overrightarrow{O M}\) with coordinates \((x_{1}, y_{1}, z_{1})\). The line passing through the point \( M \) is defined by a direction vector \(\vec{e}\), which forms angles \(\alpha, \beta, \gamma\) with the axes \(O x\), \(O y\), \(O z\). Let \(\vec{r}\) be the variable position vector corresponding to points on the considered line. Given that the vector \(\vec{r} - \vec{a}\) is collinear with the vector \(\vec{e}\), write the equation of the line passing through the given point in vector form, and then in the form of a system of equations in coordinates.

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