
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 $x^{2}+\left(a+1\right)x+4=0\left(a\in R\right)$, which has two roots $x_{1}$ and $x_{2}$, where ${x}_{1}=1+\sqrt{3}i$.  
$(1)$ Find the value of $a$;  
$(2)$ Let $x_{1}$ and $x_{2}$ correspond to points $A$ and $B$ in the complex plane, respectively. Find the length of the line segment $AB$.

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