
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 $f(x)=\log_{a}\left(b-\frac{2}{x+1}\right)$ where $(a>0$ and $a\neq 1)$ is an odd function.<br/>$(1)$ Find the solution set for the inequality $f(x) > 0$.<br/>$(2)$ Does there exist a real number $a$ such that when $x\in [m,n]$, $f(x)\in [1+\log_{a}(2n), 1+\log_{a}(2m)]$? If yes, find the range of values for $a$; if not, explain the reason.

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