
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
If the points \( A_{1}, B_{1}, C_{1} \) divide the sides of triangle \( A B C \) in such a way that
\[ \frac{A C_{1}}{B C_{1}}=\frac{B A_{1}}{C A_{1}}=\frac{C B_{1}}{A B_{1}}, \]
then the centroid of triangle \( A_{1} B_{1} C_{1} \) coincides with the centroid of the original triangle \( A B C \).

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