
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 $a > 0$, $a \neq 1$, and the function $f(x) = \log_a(x^2 - 2x + 3)$ has a maximum value. Then, the solution set of the inequality $\log_a(x-1) > 0$ is \_\_\_\_\_\_.

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