
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 $x^{2}-2(k+1)x+k^{2}+3=0$ has two distinct real roots $x_{1}$ and $x_{2}$.  
$(1)$ If $\frac{1}{{x}_{1}}+\frac{1}{{x}_{2}}=\frac{6}{7}$, find the value of $k$.  
$(2)$ Find the range of values of ${x}_{1}^{2}+{x}_{2}^{2}$.

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