
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
The real solutions of the quadratic equation $x^{2}+2x+k+1=0$ with respect to $x$ are $x_{1}$ and $x_{2}$.  
$(1)$ Find the range of values for $k$;  
$(2)$ If $x_{1}+x_{2}-x_{1}x_{2} \lt -1$ and $k$ is an integer, find the value of $k$.

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