
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
In the plane of triangle \( ABC \), perpendicular lines are drawn at each vertex, and on these lines, on the same side of the plane, the segments \( AA_{1} = k \cdot BC \), \( BB_{1} = k \cdot CA \), and \( CC_{1} = k \cdot AB \) are measured. Determine \( k \) such that the centroid of triangle \( ABC \) is equidistant from points \( A_{1}, B_{1}, \) and \( C_{1} \). What is this distance?

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