
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 $p$: $\forall x \in \mathbb{R}, x^2 - kx + 2k + 5 \geqslant 0$, and proposition $q$: the equation $\frac{x^2}{4-k} + \frac{y^2}{1-k} = 1$ represents a hyperbola with foci on the $x$-axis.

$(1)$ If proposition $q$ is true, find the range of real numbers $k$.

$(2)$ If the proposition "$p \vee q$" is true, and the proposition "$p \wedge q$" is false, find the range of real numbers $k$.

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