
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 respectively on the sides $BC$, $CA$, and $AB$ of triangle $ABC$. For these points to be collinear, it is necessary and sufficient that the following relationship holds:

$$
\frac{\overline{A C_{1}}}{\overline{C_{1} B}} \cdot \frac{\overline{B A_{1}}}{\overline{A_{1} C}} \cdot \frac{\overline{C B_{1}}}{\overline{B_{1} A}} = -1
$$

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