
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
$(1)$ Given that $\alpha$ and $\beta$ are acute angles, $\sin\alpha=\frac{{\sqrt{5}}}{5}$, $\cos\beta=\frac{{\sqrt{10}}}{{10}}$, find the value of $\alpha -\beta$;<br/>$(2)$ Given the function $f(x)={\sin^4}\frac{x}{2}+2\sin\frac{x}{2}\cos\frac{x}{2}-{\cos^4}\frac{x}{2}$, if $f(\alpha)=\frac{1}{5}$ and $\alpha\in(0,\frac{\pi}{2})$, find $\cos 2\alpha$.

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