
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 an ellipse E: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}} = 1$$ ($a > b > 0$) with a focal length of $2\sqrt{3}$, and the ellipse passes through the point $(\sqrt{3}, \frac{1}{2})$.
(Ⅰ) Find the equation of ellipse E;
(Ⅱ) Through point P$(-2, 0)$, draw two lines with slopes $k_1$ and $k_2$ respectively. These two lines intersect ellipse E at points M and N. When line MN is perpendicular to the y-axis, find the value of $k_1 \cdot k_2$.

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