
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 $p$: There exists $x_0 \in \mathbb{R}$ such that $x_{0}^{2}+2mx_{0}+2+m=0$.  
Proposition $q$: The equation $\frac {x^{2}}{1-2m} + \frac {y^{2}}{m+2} = 1$ represents a hyperbola.  
(1) If proposition $p$ is true, find the range of the real number $m$.  
(2) If proposition $q$ is true, find the range of the real number $m$.  
(3) Find the range of the real number $m$ such that the proposition "$p \vee q$" is false.

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