
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
(1) $\sin 63^{\circ}\cos 18^{\circ} + \cos 63^{\circ}\cos 108^{\circ} =$ \_\_\_\_\_\_ .

(2) The range of the function $y=2\sin^2x-3\sin x+1$, where $x \in \left[\frac{\pi}{6}, \frac{5\pi}{6}\right]$, is \_\_\_\_\_\_ .

(3) If acute angles $\alpha$ and $\beta$ satisfy $(1+\sqrt{3}\tan\alpha)(1+\sqrt{3}\tan\beta)=4$, then $\alpha+\beta=$ \_\_\_\_\_\_ .

(4) Given the function $f(x)=2\sin(\omega x+\varphi)$ ($\omega > 0$, $0 < \varphi < \frac{\pi}{2}$) is symmetric about the line $x=\frac{2\pi}{3}$ and its period is $\pi$, the correct statement is \_\_\_\_\_\_ . (Fill in the number)

- $①f(x)$ passes through the point $(0,\frac{3}{2})$;

- $②f(x)$ is monotonically decreasing on $\left[\frac{\pi}{12},\frac{2\pi}{3}\right]$;

- $③f(x)$ has a center of symmetry at $\left(\frac{5\pi}{12},0\right)$;

- $④$ Shifting the graph of $f(x)$ to the right by $|\varphi|$ units results in the graph of the function $y=2\sin{\omega x}$.

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