
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 distances from point $P$ to the two points $\left(0, -\sqrt{3}\right)$ and $\left(0, \sqrt{3}\right)$ is $4$. Let the trajectory of point $P$ be $C$, and the line $y=kx+1$ intersects $C$ at points $A$ and $B$.

(Ⅰ) Write the equation of $C$;  

(Ⅱ) If $\overrightarrow{OA} \perp \overrightarrow{OB}$, find the value of $k$.

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