
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 $f(x)=\sqrt{3}\sin^2x+\sin x \cos x-\frac{\sqrt{3}}{2}$.  
$(1)$ Find the smallest positive period of $f(x)$;  
$(2)$ Find the interval where $f(x)$ is monotonically decreasing;  
$(3)$ Let $\triangle ABC$ have sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively. If $f\left(\frac{A}{2}+\frac{\pi}{4}\right)=1$ and $a=2$, find the range of values for $b+c$.

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