
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 system $xOy$, two obtuse angles $\alpha$ and $\beta$ are formed with the $Ox$ axis as the initial side, and their terminal sides intersect the unit circle at points $A$ and $B$, respectively. It is known that the $x$-coordinates of $A$ and $B$ are $-\frac{\sqrt{2}}{10}$ and $-\frac{2\sqrt{5}}{5}$, respectively.  
(1) Find the value of $\tan(\alpha+\beta)$.  
(2) Find the value of $\alpha+2\beta$.

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