
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
For points \( A_{1}, B_{1}, C_{1} \) to lie on a straight line, it is necessary and sufficient that \( R=1 \) (see Problem 191) and that an even number (i.e., zero or two) of the points \( A_{1}, B_{1}, C_{1} \) lie on the sides of the triangle, rather than on their extensions.

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