
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 the function $f(x) = \cos \left( 2x - \frac{4\pi}{3} \right) + 2 \cos^2 x$.

(1) Find the maximum value of $f(x)$, and write out the set of $x$ values for which $f(x)$ attains its maximum;

(2) In $\triangle ABC$, let $a$, $b$, and $c$ denote the lengths of the sides opposite to angles $A$, $B$, and $C$, respectively. If $f(B+C) = \frac{3}{2}$ and $b+c=2$, find the minimum value of $a$.

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