
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 point P is any moving point on circle C. A perpendicular line to the x-axis is drawn from point P, and its foot is H. It is given that $\overrightarrow{OQ} = \frac{1}{2}(\overrightarrow{OP} + \overrightarrow{OH})$, where Q is a moving point. The trajectory of Q is E. The trajectory E intersects the positive x-axis and the positive y-axis at points A and B, respectively. The line $y=kx$ ($k>0$) intersects line AB at point D and intersects trajectory E at points M and N.  
(Ⅰ) Find the equation of trajectory E;  
(Ⅱ) Find the maximum value of the area of quadrilateral AMBN.

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