
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 proposition $p$: The graph of the equation $\frac{x^2}{4-m} + \frac{y^2}{m} = 1$ is an ellipse with foci on the $x$-axis; proposition $q$: "$\forall x \in \mathbb{R}, x^2 + 2mx + 1 > 0$"; proposition $s$: "$\exists x \in \mathbb{R}, mx^2 + 2mx + 2 - m = 0$.
$(1)$ If proposition $s$ is true, find the range of real numbers $m$;
$(2)$ If $p \lor q$ is true, and $\neg q$ is true, find the range of real numbers $m$.

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