
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 $f(x)=\sin(\frac{π}{4}+x)\sin(\frac{π}{4}-x)+\sqrt{3}\sin x\cos x$.

$(1)$ Find the value of $f(\frac{π}{6})$.

$(2)$ In acute triangle $\triangle ABC$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $f(\frac{A}{2})=1$ and $a=2$, find the range of values for $b+c$.

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