
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
Observe the following expressions and explore their patterns: 

$\frac{{x+1}}{{x-1}}=\frac{{x-1+2}}{{x-1}}=\frac{{x-1}}{{x-1}}+\frac{2}{{x-1}}=1+\frac{2}{{x-1}}$; 

$\frac{{2x-3}}{{x+1}}=\frac{{2x+2-5}}{{x+1}}=\frac{{2x+2}}{{x+1}}+\frac{{-5}}{{x+1}}=2+(-\frac{5}{{x+1}})$.

$(1)$ Fill in the blanks according to the above expressions: 

① $\frac{{3x+5}}{{x+1}}=3+$______. 

② $\frac{{ax+b}}{{x+c}}=a+$______. 

$(2)$ Find the minimum value of the fraction $\frac{{2x-8}}{{x+2}}$ where $x\geq0$. 

$(3)$ Given that $x$ is an integer, find the sum of all values of $x$ that make the value of the fraction $\frac{{{x^2}-2x}}{{x+1}}$ an integer.

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