
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 $f(x)=x^{2}+\ln x-ax$ where $a\in \mathbb{R}$.<br/>$(1)$ If $f(x)\geqslant 0$ holds for all $x\geqslant 1$, find the range of real number $a$;<br/>$(2)$ If the function $y=f(x)$ has two critical points $x_{1}$ and $x_{2}$, and $x_{1}\in (0,1)$, and $f(x_{1})-f(x_{2}) \gt m$ holds, find the range of real number $m$.

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