
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 $\alpha$ rotates counterclockwise by $\frac{\pi}{6}$ and intersects the unit circle at $(\frac{3\sqrt{10}}{10}, \frac{\sqrt{10}}{10})$, and $\tan(\alpha+\beta)=\frac{2}{5}$.
(1) Find the value of $\sin(2\alpha+\frac{\pi}{6})$,
(2) Find the value of $\tan(2\beta-\frac{\pi}{3})$.

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