
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 condition $p$: The real number $x$ satisfies $(x-a)(x-3a) < 0, a \ne 0$; condition $q$: The real number $x$ satisfies $8 < 2^{x+1} \leqslant 16$.

$(1)$ If $a=1$ and "p and q" is true, find the range of the real number $x$;

$(2)$ If $q$ is a sufficient but not necessary condition for $p$, find the range of the real number $a$.

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