
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 points $A\left(\begin{matrix} \cos \alpha , & \sin \alpha \end{matrix}\right)$ and $B\left(\begin{matrix} \cos \beta , & \sin \beta \end{matrix}\right)$, where $\alpha$ and $\beta$ are acute angles, and the distance between $A$ and $B$ is $\frac{\sqrt{10}}{5}$.

$(1)$ Find the value of $\cos (\alpha -\beta)$; $(2)$ If $\tan \frac{\alpha}{2}=\frac{1}{2}$, find the values of $\cos \alpha$ and $\cos \beta$.

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