
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 $p$: real number $x$ satisfies $(x-3a)(x-a) < 0$, where $a\neq 0$, and $q$: real number $x$ satisfies $\dfrac{x-3}{x-2}\leqslant 0$

$(1)$ If $a=1$ and $p^q$ is true, find the range of real numbers $x$.

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

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