
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 the equation of circle C is $x^2+y^2=4$, and two tangents are drawn from point M(2, 4) to circle C, touching the circle at points A and B, respectively. The line AB exactly passes through the right and top vertices of the ellipse $$T: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$$.
(1) Find the equation of the ellipse T;
(2) Given that line $l$ intersects ellipse T at two distinct points P and Q, and the equation of line $l$ is $$y=kx+ \sqrt{3} (k > 0)$$, with O being the origin, find the maximum area of triangle OPQ.

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