
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 $f\left(x\right)=a^{2}x^{3}+3ax^{2}-bx-1$ has a local extremum of $0$ at $x=1$.<br/>$(1)$ Find the values of real numbers $a$ and $b$;<br/>$(2)$ If $f\left(x\right)\geqslant m$ holds for all $x\in\left[0,+\infty \right)$, find the range of real number $m$.

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