
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 that for two quantities $A$ and $B$, if $A$ and $B$ have the same range of values, then $A$ and $B$ can be interchanged. For example, $f(x)=x+1$, $x\in[0,1]$ and $g(x)=2x-1,x\in[1, \frac{3}{2}]$, it is easy to know that $f(x)$ and $g(x)$ can be interchanged.  
$(1)$ Given $f(x)=x^{2}+bx+c$ for any $x\in\mathbb{Z}$ always has $f(x)\geqslant f(0)$, and $a=\sin \theta,\theta\in[- \frac{\pi}{2}, \frac{\pi}{2}]$, judge whether $a$ and $b$ can be interchanged.  
$(2)$ Given $f(x)= \frac{x^{2}+kx+1}{x^{2}+x+1}(x > 0)$ for any positive numbers $a$, $b$, $c$, $f(a)$, $f(b)$, $f(c)$ can form the three sides of a triangle, and $g(x)=2^{x}- \frac{3}{2},x\in[m,n]$, if $k$ and $g(x)$ can be interchanged, find the value of $m+n$.

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