
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
Let a line passing through the origin \\(O\\) intersect a circle \\((x-4)^{2}+y^{2}=16\\) at point \\(P\\), and let \\(M\\) be the midpoint of segment \\(OP\\). Establish a polar coordinate system with the origin \\(O\\) as the pole and the positive half-axis of \\(x\\) as the polar axis.  
\\((\\)Ⅰ\\()\\) Find the polar equation of the trajectory \\(C\\) of point \\(M\\);  
\\((\\)Ⅱ\\()\\) Let the polar coordinates of point \\(A\\) be \\((3, \dfrac {π}{3})\\), and point \\(B\\) lies on curve \\(C\\). Find the maximum area of \\(\\triangle OAB\\).

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