
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 the equation $x^{2}-4x+k+1=0$ has two real roots for $x$.  
$(1)$ Find the range of values for $k$;  
$(2)$ Let the two real roots of the equation be $x_{1}$ and $x_{2}$, and $\frac{3}{{x}_{1}}+\frac{3}{{x}_{2}}={x}_{1}{x}_{2}-4$. Find the value of the real number $k$.

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