
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
Given two circles C<sub>1</sub>: $x^{2}+y^{2}-2y=0$, C<sub>2</sub>: $x^{2}+(y+1)^{2}=4$ with centers C<sub>1</sub>, C<sub>2</sub> respectively, and P is a moving point such that the product of the slopes of lines PC<sub>1</sub> and PC<sub>2</sub> is $-\frac{1}{2}$.

(1) Find the equation of the trajectory M of the moving point P;  
(2) Determine whether there exists a line l passing through point A(2, 0) that intersects the trajectory M at two distinct points C and D, such that $|C_{1}C|=|C_{1}D|$. If it exists, find the equation of line l; if not, explain why.

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