
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 $\overrightarrow{a}=(\sin \alpha ,\sin \beta )$, $\overrightarrow{b}=(\cos (\alpha -\beta ),-1)$, and $\overrightarrow{c}=(\cos (\alpha +\beta ),2)$, where $\alpha, \beta \neq k\pi + \frac{\pi }{2}, k\in \mathbb{Z}$

1. If $\overrightarrow{b}$ is parallel to $\overrightarrow{c}$, find the value of $\tan \alpha \cdot \tan \beta$.
2. Find the value of ${\overrightarrow{a}}^{2}+\overrightarrow{b}\cdot \overrightarrow{c}$.

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