
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 a function $f(x)=\cos(2x+\frac{2\pi}{3})+2\cos^2x$.

(1) Find the maximum value of $f(x)$ and determine the set of values of $x$ for which $f(x)$ attains its maximum value.

(2) In triangle $ABC$, if the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is known that $f(A)=\frac{3}{2}$ and $b+c=2$, find the minimum value of $a$.

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