
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 $0＜α＜\frac{π}{3}$, $-\frac{π}{2}＜β＜-\frac{π}{3}$, $cos({α+\frac{π}{6}})=\frac{{\sqrt{3}}}{3}$, $sin({α-β+\frac{π}{6}})=\frac{2}{3}$. Find:

$(1)$ $\sin \alpha$;

$(2)$ $\cos \left(3\alpha -\beta \right)$.

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