
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 that F<sub>1</sub>(-1,0) and F<sub>2</sub>(1,0) are the left and right foci of the ellipse G: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ (where a > b > 0), let point P be on ellipse G, and let PF<sub>2</sub> be perpendicular to F<sub>1</sub>F<sub>2</sub>. Given that |PF<sub>1</sub>| - |PF<sub>2</sub>| = $$\frac{a}{2}$$:
(1) Find the equation of ellipse G.
(2) Assume point B is the top vertex of ellipse G. A straight line l passing through F<sub>2</sub> intersects the ellipse G at two distinct points M and N. Does there exist a line l such that the ratio of areas of triangles BF<sub>2</sub>M and BF<sub>2</sub>N is 2? If such a line l exists, find its equation; if not, explain why.

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