
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 $f\left(x\right)=\left(2m+1\right)x^{2}-mx+2m-1\left(m\in R\right)$.  
$(1)$ If $f\left(x\right)=\left(2m+1\right)x^{2}-mx+2m-1$ is a monotonic function on $\left[1,2\right]$, find the range of real number $m$;  
$(2)$ If $m\leqslant -1$, solve the inequality with respect to $x$: $f\left(x\right)\geqslant 2m+mx^{2}$.

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