
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 the cosine formula for the sum of two angles $C_{(\alpha+\beta)}$: $\cos(\alpha-\beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$;  
1. Derive the sine formula for the sum of two angles $S_{(\alpha+\beta)}$: $\sin(\alpha+\beta) = \_\_\_\_\_\_$  
2. Given $\cos\alpha = -\frac{4}{5}$, $\alpha \in (\pi, \frac{3}{2}\pi)$, $\tan\beta = -\frac{1}{3}$, $\beta \in (\frac{\pi}{2}, \pi)$, find $\sin(\alpha+\beta)$.

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