
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
Proposition $P$: The function $y=\lg (-x^{2}+4ax-3a^{2})$ ($a > 0$) is meaningful. Proposition $q$: $\left\{x|y= \sqrt{-{x}^{2}+11x-18}\right\}$.  
$(1)$ When $a=1$ and $p\land q$ is true, find the range of real numbers $x$.  
$(2)$ If $\lnot p$ is a necessary but not sufficient condition for $\lnot q$, find the range of real numbers $a$.

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