
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 $A \begin{pmatrix} \cos \alpha, \sin \alpha \end{pmatrix}$ and $B \begin{pmatrix} \cos \beta, \sin \beta \end{pmatrix}$, where $\alpha$ and $\beta$ are acute angles, and $|AB| = \frac{\sqrt{10}}{5}$.  
$(1)$ 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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