
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 $a > 0$, $a \neq 1$ and $\log_a 3 > \log_a 2$, if the function $f(x) = \log_a x$ has a difference of 1 between its maximum and minimum values in the interval $[a, 2a]$, then:  
(1) Find the value of $a$;  
(2) Solve the inequality $\log_{\frac{1}{3}}(x-1) > \log_{\frac{1}{3}}(a-x)$;  
(3) Determine the intervals of monotonicity for the function $g(x) = |\log_a x - 1|$.

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