
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 that $x_{1}$ and $x_{2}$ are two real roots of the one-variable quadratic equation $4kx^{2}-4kx+k+1=0$.  
$(1)$ Find the range of real numbers for $k$;  
$(2)$ If $x_1^2+x_2^2=4$, find the value of $k$;  
$(3)$ Find the integer values of the real number $k$ that make the value of $\frac{x_1}{x_2}+\frac{x_2}{x_1}-2$ an integer.

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