
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: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a > b > 0)$$ where the major axis is a diameter of the circle $x^2+y^2=4$, and the distance from the right focus to the line $x+y-2\sqrt{3}=0$ is $\frac{\sqrt{6}}{2}$.

(1) Find the standard equation of the ellipse C.
(2) Determine whether there exists a line $l: y=kx+m$ $(k\in \mathbb{R})$ that intersects the ellipse C at points A and B in such a way that $$|2\vec{OA} + \vec{OB}| = |2\vec{OA} - \vec{OB}|$$ holds. If such a line exists, find the range of values for the real number $m$; if not, provide an explanation.

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