
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 the domain of the function $y=\lg (-x^{2}+5x-4)$ be set $A$, and the range of the function $y=\frac{3}{x+1}$, $x\in \left(0,m\right)$ be set $B$.  
$(1)$ When $m=1$, find $A\cup B$;  
$(2)$ If "$x\in A$" is a necessary but not sufficient condition for "$x\in B$", find the range of the real number $m$.

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