
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 $\sin \alpha + \cos \alpha = \frac{3 \sqrt{5}}{5}$, where $\alpha \in \left(\begin{matrix} \begin{matrix}0, \frac{\pi}{4} \end{matrix}\end{matrix}\right)$, and $\sin \left(\begin{matrix} \begin{matrix}\beta - \frac{\pi}{4} \end{matrix}\end{matrix}\right) = \frac{3}{5}$, where $\beta \in \left(\begin{matrix} \begin{matrix} \frac{\pi}{4}, \frac{\pi}{2} \end{matrix}\end{matrix}\right)$:

(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}
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