
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
In the plane rectangular coordinate system $xOy$, for point $A(a,b)$, if the function $y=f(x)$ satisfies: for all $x\in [a-1,a+1]$, there exists $y\in [b-1,b+1]$, then the function is called a "$1$-boundary function" of point $A$.  
$(1)$ If the function $y=x$ is a "$1$-boundary function" of point $A(a,b)$, find the relationship that $a$ and $b$ need to satisfy.  
$(2)$ If point $B(m,n)$ lies on the graph of the function $y=-\frac{1}{2}x^2$, does there exist an $m$ such that the function $y=-\frac{1}{2}x^2$ is a "$1$-boundary function" of point $B$? If it exists, find the range of values for $m$; if it does not exist, explain the reason.

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