
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 $\dfrac{\pi}{6}$ and intersects the unit circle at point $\left( \dfrac{3\sqrt{10}}{10}, \dfrac{\sqrt{10}}{10} \right)$, and $\tan (α+β)=\dfrac{2}{5}$.

$(1)$ Find the value of $\sin \left( 2\alpha + \dfrac{\pi}{6} \right)$;

$(2)$ Find the value of $\tan \left( 2\beta - \dfrac{\pi}{3} \right)$.

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