
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 $(0, -\sqrt{3})$ and $(0, \sqrt{3})$ equals 4. Let the trajectory of point P be C.  
(1) Write the equation of C;  
(2) Suppose the line $y=kx+1$ intersects C at points A and B. For what value of $k$ is $\overrightarrow{OA} \perp \overrightarrow{OB}$?

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