
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 \\(f(x)=\cos^2x- \frac{\sqrt{3}}{2}\sin 2x- \frac{1}{2}\\),  
\\((1)\\) find the coordinates of the symmetry center of the graph of \\(f(x)\\);  
\\((2)\\) In triangle \\(ABC\\), the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) are \\(a\\), \\(b\\), and \\(c\\) respectively. If \\(f(A)+1=0\\) and \\(b+c=2\\), find the minimum value of \\(a\\).

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