
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 $\vec{a} = (\cos \alpha, \sin \alpha), \vec{b} = (\cos \beta, \sin \beta)$, and $\left| \vec{a} - \vec{b} \right| = \sqrt{2}$.
1. Find the value of $\cos(\alpha - \beta)$;
2. If $\pi < \beta < 0 < \alpha < 2\pi$, and $\sin \beta = -\frac{\sqrt{2}}{2}$, find the value of $\sin \alpha$.

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