
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 $f(x) = A\sin(\omega x + \frac{\pi}{6})$ ($A > 0$, $\omega > 0$), the coordinates of the first highest point and the first lowest point on the right side of the y-axis are $(x_0, 2)$ and $(x_0 + \frac{\pi}{2}, -2)$, respectively.  
(1) Find the analytical expression of the function $f(x)$.  
(2) In $\triangle ABC$, the sides opposite angles A, B, and C are denoted as $a$, $b$, and $c$, respectively, and it is given that $2\sin A\sin C + \cos 2B = 1$. Find the range of values for $g(B) = \sqrt{3}f(B) + f(B + \frac{\pi}{4})$.

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