
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 vectors $\overrightarrow{a} = (\cos \alpha, \sin \alpha)$ and $\overrightarrow{b} = (\cos \beta, \sin \beta)$, with $|\overrightarrow{a} - \overrightarrow{b}| = \frac{2\sqrt{5}}{5}$.

(Ⅰ) Find the value of $\cos (\alpha - \beta)$;

(Ⅱ) If $0 < \alpha < \frac{\pi}{2}$ and $-\frac{\pi}{2} < \beta < 0$, and $\sin \beta = -\frac{5}{13}$, find the value of $\sin \alpha$.

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