
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 $p$: for real number $x$, $x^{2}-2ax-3a^{2} \lt 0\left(a \gt 0\right)$, $q: 2\leqslant x \lt 4$.  
$(1)$ If $a=1$, and both $p$ and $q$ are true propositions, find the range of values for $x$.  
$(2)$ If $q$ is a sufficient but not necessary condition for $p$, find the range of real numbers for $a$.

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