
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 lines are drawn through the vertices \( A, B, C \) of triangle \( ABC \) and an arbitrary point \( O \) in its plane intersecting the sides \( AB, AC, BC \) respectively at points \( C_1, B_1 \), and \( A_1 \), defining two segments on each side, then the products of any three segments that do not have a common vertex are equal: \( AC_1 \cdot BA_1 \cdot CB_1 = AB_1 \cdot CA_1 \cdot BC_1 \).

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