
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 proposition $p$: For real number $x$, $x^{2}-4ax+3a^{2} \lt 0$ (where $a \gt 0$), proposition $q$: For real number $x$, $\left\{\begin{array}{l}|{x-1}|≤2\\ \frac{{x+3}}{{x-2}}≥0\end{array}\right.$.<br/>$(1)$ If $a=1$, and propositions $p$ and $q$ are both true, find the range of real number $x$; <br/>$(2)$ If $q$ is a sufficient but not necessary condition for $p$, find the range of real number $a$.

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