
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 $$C: \frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ and a hyperbola $$\frac {x^{2}}{4-v}+ \frac {y^{2}}{1-v}=1 (1<v<4)$$ share a common focus. A line $l$ passes through the right vertex B of the ellipse C and intersects the parabola $y^2=2x$ at points P and Q, with $OP \perpendicular OQ$.

(Ⅰ) Find the equation of the ellipse C;

(Ⅱ) On the ellipse C, is there a point R $(m, n)$ such that the line $l: mx+ny=1$ intersects the circle $O: x^2+y^2=1$ at two distinct points M and N, and the area of $\triangle OMN$ is maximized? If such a point exists, find the coordinates of point R and the corresponding area of $\triangle OMN$; if not, explain why.

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