
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 $p$: $\frac{1}{x} \lt 1$, $q: x^{2}-3ax+2a^{2}\leqslant 0$ (where $a$ is a constant, and $a \gt 0$).  
$(1)$ If $p$ is true, find the range of $x$.  
$(2)$ If $p$ is a necessary but not sufficient condition for $q$, find the range of $a$.

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