
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 $f(x) = \log_a x$, its inverse function is $g(x)$.  
(1) Solve the equation for $x$: $f(x-1) = f(a-x) - f(5-x)$;  
(2) Let $F(x) = (2m-1)g(x) + \left( \frac{1}{m} - \frac{1}{2} \right)g(-x)$. If $F(x)$ has a minimum value, find the expression for $h(m)$;  
(3) Find the maximum value of $h(m)$.

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