
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 $m\in R$, $p$: "The domain of the function $f\left(x\right)=\ln (mx^{2}-mx+1)$ is $R$", $q$: "$\exists x_{0}\in \left[0,3\right]$, such that $x_0^2-2{x_0}-m≥0$ holds".  
$(1)$ If $q$ is a true proposition, find the range of real numbers for $m$;  
$(2)$ If "$p\vee q$" is a true proposition and "$p\wedge q$" is a false proposition, find the range of real numbers for $m$.

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