
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 $\cos(\alpha + \beta) = \frac{2\sqrt{5}}{5}$, $\tan \beta = \frac{1}{7}$, and $\alpha, \beta \in \left(0, \frac{\pi}{2}\right)$.  
$(1)$ Find the value of $\cos 2\beta + \sin 2\beta - \sin \beta \cos \beta$.  
$(2)$ Find the value of $2\alpha + \beta$.

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