
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 one-variable quadratic equation about $x$ is $x^{2}+\left(2k+1\right)x+k^{2}+1=0$ with two distinct real roots $x_{1}$ and $x_{2}$.  
$(1)$ Find the range of real number $k$.  
$(2)$ If the two real roots $x_{1}$ and $x_{2}$ of the equation satisfy $|x_{1}|+|x_{2}|=x_{1}\cdot x_{2}$, find the value of $k$.

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