
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 $f(x)=2\cos^2x+\sin(2x- \frac{\pi}{6})$  
$(1)$ Find the intervals of increase for the function $f(x)$; its maximum value, and the set of values of $x$ for which the maximum value is attained;  
$(2)$ Given in $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. If $f(A)= \frac{3}{2}$ and $b+c=2$, find the range of the real number $a$.

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