
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 $\sin \alpha = \frac{\sqrt{5}}{5}$, $\sin \left( \alpha - \beta \right) = -\frac{\sqrt{10}}{10}$, and both $\alpha$ and $\beta$ are acute angles, find the value of $\cos 2\beta$. The options are:
A) $-\frac{\sqrt{3}}{2}$
B) $-1$
C) $0$
D) $1$

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