
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 $p$: $\forall x \in \mathbb{R}$, $x^{2}-ax+a > 0$;  
$q$: The equation $\dfrac {x^{2}}{a^{2}+12}- \dfrac {y^{2}}{4-a^{2}}=1$ represents a hyperbola.  
$(1)$ If $p$ is a true proposition, find the range of real numbers $a$;  
$(2)$ When $p$ is a false proposition, and $q$ is a true proposition, find the range of real numbers $a$.

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