
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 $p$: real number $x$ satisfies $x^{2}-4mx+3m^{2} \lt 0$; proposition $q$: real number $x$ satisfies $|x-3|\leqslant 1$.  
$(1)$ If $m=1$, and both $p$ and $q$ are true, find the range of real number $x$;  
$(2)$ If $m \gt 0$, and $q$ is a sufficient but not necessary condition for $p$, find the range of real number $m$.

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