
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 $α∈({0,\frac{π}{4}})$, and $sinα+cosα=\frac{{2\sqrt{10}}}{5}$.  
$(Ⅰ)$ Find the value of $\tan \alpha$;  
$(Ⅱ)$ If $β∈({0,π})$, and $\tan({2α+β})=-\frac{1}{2}$, find the value of $\alpha +\beta$.

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