
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 two points $(-\sqrt{3},0)$ and $(\sqrt{3},0)$ is equal to $4$. Let the trajectory of point $P$ be curve $C$.

(I) Write the equation of $C$.

(II) If the line $y=k(x+1)$ intersects $C$ at points $A$ and $B$, what value of $k$ makes $\overrightarrow{OA} \perp \overrightarrow{OB}$? What is the value of $|\overrightarrow{AB}|$ at this time?

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