
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 $p: x^{2}-6ax+8a^{2} \lt 0$ ($a\neq 0$), $q: x^{2}-4x+3\leqslant 0$.  
$(1)$ When $a=1$, if both $p$ and $q$ hold true, find the range of real number $x$.  
$(2)$ If $p$ is a sufficient but not necessary condition for $q$, find the range of real number $a$.

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