
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
Let \( A_{1}, B_{1}, \) and \( C_{1} \) be three points lying on the sides \( BC, CA, \) and \( AB \) of triangle \( ABC \), respectively. The lines \( AA_{1}, BB_{1}, \) and \( CC_{1} \) intersect at a single point, or are all parallel, if and only if the following relationship holds:

\[ 
\frac{\overline{A_{C1}}}{\overline{C_{1}B}} \cdot \frac{\overline{B A_{1}}}{\overline{A_{1}C}} \cdot \frac{\overline{C B_{1}}}{\overline{B_{1}A}} = 1 
\]

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