
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) Given that $\sin(\frac{\pi}{3}-\alpha)=\frac{1}{6}$, find the value of $\cos(\frac{\pi}{6}+\alpha)$.
(2) Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two non-collinear vectors, $\overrightarrow{AB}=2\overrightarrow{a}+k\overrightarrow{b}$, $\overrightarrow{BC}=\overrightarrow{a}+\overrightarrow{b}$, and $\overrightarrow{CD}=\overrightarrow{a}-2\overrightarrow{b}$. If points $A$, $B$, and $D$ are collinear, find the value of the real number $k$.
(3) Given that $\alpha=-\frac{35}{6}\pi$, find the value of $\frac{2\sin(\pi+\alpha)\cos(\pi-\alpha)-\sin(\frac{3\pi}{2}+\alpha)}{1+\sin^{2}\alpha-\cos(\frac{\pi}{2}+\alpha)-\cos^{2}(\pi+\alpha)}$.
(4) If $\sin\theta+\cos\theta=\frac{17}{13}$ and $\theta\in(0,\frac{\pi}{4})$, find the value of $\tan\theta$.
(5) If the function $f(x)=\sin(\omega x+\frac{\pi}{3})$ ($\omega > 0$) attains its unique maximum value of 1 and minimum value of -1 on the interval $[0,2\pi]$, find the range of $\omega$.
(6) If the graph of the function $f(x)=3\sin(2x-\frac{\pi}{3})$ is denoted as $C$, determine the correct conclusion(s) from the following options:
   (①) Graph $C$ is symmetric with respect to the line $x=\frac{11\pi}{12}$.
   (②) Graph $C$ is symmetric with respect to the point $(\frac{2\pi}{3},0)$.
   (③) Function $f(x)$ is not a monotonic function on the interval $(-\frac{\pi}{12},\frac{5\pi}{12})$.
   (④) The graph of $y=3\sin 2x$ can be translated to the right by $\frac{\pi}{3}$ units to obtain graph $C$.

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