
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 that the terminal side of angle $α$ rotates counterclockwise by $\dfrac{π}{6}$ and intersects the unit circle at the point $\left( \dfrac{3 \sqrt{10}}{10}, \dfrac{\sqrt{10}}{10} \right)$, and $\tan (α+β)= \dfrac{2}{5}$.
$(1)$ Find the value of $\sin (2α+ \dfrac{π}{6})$,
$(2)$ Find the value of $\tan (2β- \dfrac{π}{3})$.

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