
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 the two real roots of the one-variable quadratic equation $x^{2}-2\left(k+1\right)x+k^{2}+2=0$. Find:<br/>
$(1)$ $x_{1}+x_{2}=$______；$x_{1}x_{2}=$______；<br/>
$(2)$ If $(x_{1}+1)(x_{2}+1)=8$, find the value of $k$.

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