
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 the function $f(x)=\log _{a}(2x-4)+\log _{a}(5-x)$ $(a \gt 0$, and $a\neq 1)$, and its graph passes through the point $P(3,-2)$.  
$(1)$ Find the value of $a$ and the domain of $f(x)$;  
$(2)$ Find the minimum value of $f(x)$ on the interval $[3,\frac{9}{2}]$;  
$(3)$ If ${2}^{m}={3}^{n}=t$ $(\frac{5}{2}＜t＜3)$, compare the sizes of $f(2m)$ and $f(3n)$.

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