
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 that $f(x)$ is an odd function defined on $[-1,1]$ and $f(1)=1$. If $a$, $b \in [-1,1]$ and $a+b \neq 0$, then $\frac{f(a)+f(b)}{a+b} > 0$.

$(1)$ Determine the monotonicity of the function $f(x)$ on $[-1,1]$;

$(2)$ If $f(x) \leqslant m^{2}-2am+1$ holds for all $x \in [-1,1]$, find the range of the real number $m$.

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