
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
In the Cartesian coordinate plane $(xOy)$, the sum of the distances from point $P$ to points $F\_1(0,-\sqrt{3})$ and $F\_2(0,\sqrt{3})$ is equal to $4$. Let the trajectory of point $P$ be denoted as $C$.
(1) Find the equation of trajectory $C$;
(2) Let line $l$: $y=kx+1$ intersect with curve $C$ at points $A$ and $B$. For what value of $k$ does $| \overrightarrow{OA}+ \overrightarrow{OB}|=| \overrightarrow{AB}|$ hold true (where $O$ is the origin of the coordinate plane)? What is the value of $| \overrightarrow{AB}|$ in this case?

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