
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 quadratic equation $4kx^{2}-4kx+k+1=0$.  
$(1)$ Is there a real number $k$ such that $(2x_{1}-x_{2})(x_{1}-2x_{2})=-\frac{3}{2}$ holds true? If yes, find the value of $k$; if not, explain the reason.  
$(2)$ Find the integer values of the real number $k$ that make $\frac{{x}_{1}}{{x}_{2}}+\frac{{x}_{2}}{{x}_{1}}-2$ an integer.

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