
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
Let the function $f(x)=\cos (2x- \frac {4\pi}{3})+2\cos ^{2}x$.  
(I) Find the maximum value of $f(x)$, and write the set of $x$ values that make $f(x)$ take its maximum value;  
(II) Given in $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$. If $f(B+C)= \frac {3}{2},b+c=2.$ Find the minimum value of $a$.

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