
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 $f(x)=\frac{mx+n}{{x}^{2}+1}$ defined on $\left[-1,1\right]$ is an odd function, and $f\left(1\right)=1$.  
$(1)$ Find the values of $m$ and $n$;  
$(2)$ Find the range of real numbers $a$ that satisfy $f\left(a-1\right)+f(a^{2}-1) \lt 0$.

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