
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 $m=2\log _{3}x$ and $n=\log _{5}(x-1)$.  
$(1)$ When $3^{m}-5^{n}=3$, find the value of $x$.  
$(2)$ When $x=4$, express $\log _{15}20$ in terms of $m$ and $n$.

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