
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 $f(x) = \cos^2x - \frac{\sqrt{3}}{2}\sin2x - \frac{1}{2}$,  
(1) Write down the coordinates of the symmetry center of the graph of $f(x)$ and the interval where $f(x)$ is monotonically increasing;  
(2) In triangle $\triangle ABC$, 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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