
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
Consider three circles with centers $A(0, \sqrt{3})$, $B(-2, 0)$, and $C(2, 0)$ each having radius $r = \sqrt{5}$. 
If $D$ is the point of intersection of the circle centered at $A$ with the circle centered at $B$ that is outside the circle centered at $C$, and if $E$ is the point of intersection of the circle centered at $A$ with the circle centered at $C$ that is outside the circle centered at $B$, calculate the length of the line segment $DE$.
A) $1$
B) $2$
C) $3$
D) $4$
E) $5$

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