
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 points A$(x_0, 0)$ and B$(0, y_0)$ moving respectively on the x-axis and y-axis, and $|AB|=1$. If the moving point P$(x, y)$ satisfies $$\overrightarrow{OP} = 2\overrightarrow{OA} + \sqrt{3}\overrightarrow{OB}$$.
(I) Find the standard equation of the curve C, which is the trajectory of the moving point P;
(II) A line $l_1$ with a vertical intercept of 2 intersects curve C at points P and Q. If the circle with diameter PQ passes exactly through the origin, find the equation of the line;
(III) The line $l_2$: $x=ty+1$ intersects curve C at points A and B, with E$(1, 0)$. When t varies, is there a line $l_2$ such that the area of $\triangle ABE$ is $2\sqrt{3}$? If it exists, find the equation of the line $l_2$; if not, explain why.

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