
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 an acute-angled \\(\triangle ABC\\) with the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) being \\(a\\), \\(b\\), and \\(c\\) respectively, and \\(\dfrac{b}{2}\\) is the arithmetic mean of \\(2a\sin A\cos C\\) and \\(c\sin 2A\\).  
\\((\\)I\\()\\) Find the magnitude of angle \\(A\\);  
\\((\\)II\\()\\) If \\(a=2\\), find the maximum area of \\(\triangle ABC\\).

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