
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 a quadratic function $y=-x^{2}+4x+m-1$.  
$(1)$ When $m=3$,  
① Find the coordinates of the vertex of the function;  
② When $n\leqslant x\leqslant n+2$, the maximum value of the function is $3$, find the value of $n$;  
$(2)$ If there are exactly $2$ points on the graph of the function that are $3$ units away from the $x$-axis, directly write down the range of values for $m$.

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