
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 real numbers \(a\) and \(b\), define the function \(f(x) = \frac{1}{ax+b}\). For which values of \(a\) and \(b\) are there distinct real numbers \(x_1\), \(x_2\), and \(x_3\) such that \(f(x_1) = x_2\), \(f(x_2) = x_3\), and \(f(x_3) = x_1\)?

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