
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 sets $A=\{x|x^{2}-4ax+3a^{2} \lt 0\}$ and $B=\{x|(x-3)(2-x)\geqslant 0\}$.

(1) When $a=1$, find $A\cap B$ and $A\cup B$;

(2) Let $a \gt 0$, if "$x\in A$" is a necessary but not sufficient condition for "$x\in B$", find the range of real numbers for $a$.

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