
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
Let proposition $p$: $\exists x_{0} \in \mathbb{R}$, $x_{0}^{2}+2mx_{0}+2+m=0$,  
proposition $q$: The equation $\dfrac {x^{2}}{1-2m}+ \dfrac {y^{2}}{m+2}=1$ represents a hyperbola  
$(1)$ If proposition $p$ is true, find the range of real numbers $m$;  
$(2)$ If proposition $q$ is true, find the range of real numbers $m$;  
$(3)$ Find the range of real numbers $m$ that makes "$p \lor q$" a false proposition.

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