
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 centers of two circles $C_1$ and $C_2$ is equal to 4, where $C_1: x^2+y^2-2\sqrt{3}y+2=0$, $C_2: x^2+y^2+2\sqrt{3}y-3=0$. Let the trajectory of point $P$ be $C$.
(1) Find the equation of $C$;
(2) Suppose the line $y=kx+1$ intersects $C$ at points $A$ and $B$. What is the value of $k$ when $\overrightarrow{OA} \perp \overrightarrow{OB}$? What is the value of $|\overrightarrow{AB}|$ at this time?

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