Given $\cos (\alpha + \beta) + \sin (\alpha - \beta) = 0$ and $\tan \beta = \frac{1}{2000},$ find $\tan \alpha.$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Given $\cos (\alpha + \beta) + \sin (\alpha - \beta) = 0$ and $\tan \beta = \frac{1}{2000},$ find $\tan \alpha.$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
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Given $\cos (\alpha + \beta) + \sin (\alpha - \beta) = 0$ and $\tan \beta = \frac{1}{2000},$ find $\tan \alpha.$
train