
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 that $a > 0$, $a \neq 1$, and $\log_a 3 > \log_a 2$, where the function $f(x) = \log_a x$, find the difference between the maximum and minimum value on the interval $[a, 2a]$ to be 1.
1. Find the value of $a$.
2. Solve the inequality $\log_{\frac{1}{2}}(x-1) > \log_{\frac{1}{2}}(a-x)$.
3. Determine the monotonic intervals of the function $g(x) = |\log_a x - 1|$.

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