
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 $\sin\alpha + \cos\alpha = \frac{3\sqrt{5}}{5}$, with $\alpha \in \left(0, \frac{\pi}{4}\right)$, and $\sin\left(\beta - \frac{\pi}{4}\right) = \frac{3}{5}$, with $\beta \in \left(\frac{\pi}{4}, \frac{\pi}{2}\right)$.  
(1) Find the values of $\sin 2\alpha$ and $\tan 2\alpha$;  
(2) Find the value of $\cos(\alpha + 2\beta)$.

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