
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 $\alpha $ and $\beta \in \left(0,\pi \right)$, and $\cos \alpha =\frac{{\sqrt{5}}}{5}$, $\sin \left(\alpha +\beta \right)=-\frac{{\sqrt{2}}}{{10}}$.  
$(1)$ Find $\cos 2\alpha $ and $\sin 2\alpha$;  
$(2)$ Find the value of $3\alpha +\beta$.

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