
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 a moving point P(x, y) and an ellipse $\frac {x^{2}}{4}+ \frac {y^{2}}{3}=1$, the product of the slopes of the lines connecting P with the two foci of the ellipse, F<sub>1</sub> and F<sub>2</sub>, is equal to a constant $\lambda$ ($\lambda \neq 0$).  
(1) Find the equation of the trajectory C of the moving point P;  
(2) Discuss the shape of C based on the values of $\lambda$.

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