
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 a function $f(x)$ defined on the interval $(0,+\infty)$ which is increasing, $f(1)=0$, and $f(3)=1$.  
$(1)$ Solve the inequality $0 \lt f(x^{2}-1) \lt 1$;  
$(2)$ If $f(x)\leq m^{2}-2am+1$ holds for all $x\in (0,3]$, $a\in [-1,1]$, find the range of real number $m$.

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