
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 $\tan \alpha =\frac{1}{3}$ and $\cos \beta =\frac{{\sqrt{5}}}{5}$, where $0 \lt \alpha\ \ \lt \frac{π}{2}$ and $\frac{{3π}}{2} \lt \beta\ \ \lt 2\pi $. Find:<br/>
$(1)$ the value of $\tan 2\alpha$;<br/>
$(2)$ the value of $\alpha +\beta$.

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