
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]


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In the Cartesian coordinate system $(xOy)$, the vertex of angle $\alpha$ is at the origin, and its initial side lies on the positive half of the $x$-axis.
(1) When the terminal side of angle $\alpha$ is the ray $l$: $y=2\sqrt{2}x$ $(x\geqslant 0)$, find the value of $\cos(\alpha+\frac{\pi}{6})$;
(2) Given that $\frac{\pi}{6}\leqslant\alpha\leqslant\frac{3\pi}{4}$, find the range of values for $\frac{3}{2}\sin(2\alpha)+\sqrt{3}\cos^2(\alpha)-\frac{\sqrt{3}}{2}$.

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