
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
For the quadratic equation in $x$: $x^{2}+\left(2k+1\right)x+k^{2}+1=0$<br/>$(1)$ If the equation has two distinct real roots, find the range of $k$;<br/>$(2)$ If $x_{1}$ and $x_{2}$ are the two roots of the equation, and $x_{1}+x_{2}=2-x_{1}\cdot x_{2}$, find the value of $k$.

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