
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 $p:x^{2}-5ax+4a^{2} \lt 0$, where $a \gt 0$, $q:3 \lt x\leqslant 4$.  
$(1)$ If $a=1$ and $p\wedge q$ is true, find the range of real number $x$.  
$(2)$ If $p$ is a necessary but not sufficient condition for $q$, find the range of real number $a$.

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