
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
Given \( A=\left\{x \mid \log _{3}\left(x^{2}-2 x\right) \leqslant 1\right\}, B=(-\infty, a] \cup(b,+\infty) \), where \( a < b \), if \( A \cup B=\mathbf{R} \), what is the minimum value of \( a - b \) ?

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