
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 $f(α)=\frac{{\sin(π+α)}}{{\tan(π-α)}}$.  
$(1)$ Find the value of $f({\frac{{5π}}{6}})$;  
$(2)$ If $f({\frac{π}{3}-α})=\frac{2}{3}$, find the value of $\sin({α+\frac{π}{6}})$.

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