
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
A rhombus is formed by connecting the four vertices of the ellipse $$C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1$$ (a>b>0) in order. The rhombus has a side length of $$\sqrt{3}$$ and an area of $$2\sqrt{2}$$.

1. Find the equation of the ellipse C.
2. A and B are two different points on the ellipse C. The product of the slopes of the lines OA and OB (where O is the origin) is $$- \frac{1}{2}$$. There is a point M on the line segment OA such that $$\frac{|OM|}{|OA|}= \frac{2}{3}$$. The line segment BM is connected and extended to intersect the ellipse C at point N. Find the value of $$\frac{|BM|}{|BN|}$$.

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