
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
Evaluate the following expressions:
1. $\sin (-1^\circ 200^\circ)\cos (1^\circ 290^\circ)+\cos (-1^\circ 020^\circ)\sin (-1^\circ 050^\circ)$
2. Given $f(\alpha)=\frac{2\sin (\pi+\alpha)\cos (3\pi-\alpha)+\cos (4\pi-\alpha)}{1+\sin ^{2}\alpha +\cos ( \frac{3\pi}{2}+\alpha )-\sin ^{2}( \frac{\pi}{2}+\alpha )}\ (1+2\sin ^{2}\alpha\neq 0)$, find $f(- \frac{23\pi}{6})$.

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