
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 two points $P(x_1,y_1)$ and $Q(x_2,y_2)$ on the function $f(x)=\sin(\omega x+\varphi)$ where $\omega > 0$ and $0 < \varphi < \frac{\pi}{2}$, if $|y_1-y_2|=2$, then the minimum value of $|x_1-x_2|$ is $\frac{\pi}{2}$. The function's graph passes through the point $(0,\frac{1}{2})$. In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $2\sin A\sin C+\cos 2B=1$.

(1) Find the analytical expression of the function $f(x)$;
(2) Find the range of values of $g(B)=\sqrt{3}f(B)+f(B+\frac{\pi}{4})$.

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