
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
Define: a line that does not coincide with the coordinate axis intersects the coordinate axis at points $A$ and $B$ (where $A$ and $B$ are not coincident). If a parabola $L$ passes through points $A$ and $B$, then this parabola is called the "friendship line" of the line.  
$(1)$ If the parabola $L$ is the "friendship line" of the line $y=-x+3$ and passes through the point $(-1,0)$, find the analytical expression of this parabola.  
$(2)$ Given that the "friendship line" of the line $y=kx+b$ is $y=-\frac{1}{2}x^{2}+\frac{1}{2}x+1$, and the line intersects the hyperbola $y=\frac{2}{x}$ at points $M$ and $N$, find the length of segment $MN$.  
$(3)$ If there is a line $y=mx+n$ and $m+n=1$, and for any real number $a$, there always exists a parabola $L$ with the "friendship line" as $y=ax^{2}+bx+c$, find the range of values for $b$.

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