
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
In the Cartesian coordinate plane $(xOy)$, two acute angles $\alpha$ and $\beta$ are formed with the non-negative half of the $x$-axis as the initial side. The terminal sides of these angles intersect the unit circle at points $A$ and $B$, respectively. Given that the $x$-coordinate of point $A$ is $\frac{\sqrt{5}}{5}$ and the $y$-coordinate of point $B$ is $\frac{\sqrt{2}}{10}$. Find:

$(1)$ the value of $\tan(\alpha + \beta)$;
$(2)$ the value of $2\alpha + \beta$.

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