
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
In the Cartesian coordinate system xOy, a moving point M has a distance ratio to the fixed points A(1,0) and B(4,0) of $\frac{1}{2}$. Let the trajectory of point M be curve C.  
(Ⅰ) Find the equation of curve C;  
(Ⅱ) A line l with a slope of 0 passes through point B and intersects curve C at points P and Q;  
(ⅰ) If OP⊥OQ, find the equation of line l;  
(ⅱ) Find the maximum area of triangle APQ.

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