
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 $p$: $\exists x_0 \in \mathbb{R}$, 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 real numbers $m$;

(2) If proposition $q$ is true, find the range of real numbers $m$;

(3) Find the range of real numbers $m$ for which “$p \lor q$” is a false proposition.

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