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

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