
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 system $xOy$, the sum of the distances from point $P$ to the points $\left(0,-\sqrt{3}\right)$ and $\left(0,\sqrt{3}\right)$ is equal to $4$. Let the locus of point $P$ be denoted as $C$.
- Write down the equation of $C$.
- Let the line $y=kx+1$ intersect $C$ at points $A$ and $B$. For what value of $k$ are the vectors $\overrightarrow {OA}$ and $\overrightarrow {OB}$ perpendicular? What is the value of $|\overrightarrow {AB}|$ in this case?

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