
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
The ellipse C is defined by the equation $$\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1(a > b > 0)$$ with left and right focal points F₁ and F₂ respectively. A is a moving point on the ellipse (distinct from the left and right vertices) such that the perimeter of △AF₁F₂ is $$4+2\sqrt{3}$$ and its maximum area is $$\sqrt{3}$$.

(I) Find the equation of the ellipse C.

(II) Let A and B be two moving points on the ellipse C, with P being the midpoint of the line segment AB. The slopes of OA and OB are k₁ and k₂ respectively (with O being the origin), and $${{k}_{1}}{{k}_{2}}=-\frac{1}{4}$$. Find the range of |OP|.

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