solution: We have that
  \begin{align*}
  \tan 100^\circ + 4 \sin 100^\circ &= \frac{\sin 100^\circ}{\cos 100^\circ} + 4 \sin 100^\circ \\
  &= \frac{\sin 80^\circ}{-\cos 80^\circ} + 4 \sin 80^\circ \\
  &= -\frac{\cos 10^\circ}{\sin 10^\circ} + 4 \cos 10^\circ \\
  &= \frac{4 \cos 10^\circ \sin 10^\circ - \cos 10^\circ}{\sin 10^\circ}.
  \end{align*}By double angle formula,
  \begin{align*}
  \frac{4 \cos 10^\circ \sin 10^\circ - \cos 10^\circ}{\sin 10^\circ} &= \frac{2 \sin 20^\circ - \cos 10^\circ}{\sin 10^\circ} \\
  &= \frac{\sin 20^\circ + \sin 20^\circ - \sin 80^\circ}{\sin 10^\circ}.
  \end{align*}By sum-to-product,
  \[\sin 20^\circ - \sin 80^\circ = 2 \cos 50^\circ \sin (-30^\circ) = -\cos 50^\circ,\]so
  \begin{align*}
  \frac{\sin 20^\circ + \sin 20^\circ - \sin 80^\circ}{\sin 10^\circ} &= \frac{\sin 20^\circ - \cos 50^\circ}{\sin 10^\circ} \\
  &= \frac{\sin 20^\circ - \sin 40^\circ}{\sin 10^\circ}.
  \end{align*}By sum-to-product,
  \[\sin 20^\circ - \sin 40^\circ = 2 \cos 30^\circ \sin (-10^\circ) = -\sqrt{3} \sin 10^\circ,\]so $\frac{\sin 20^\circ - \sin 40^\circ}{\sin 10^\circ} = \boxed{-\sqrt{3}}.$
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  Simplify $\tan 100^\circ + 4 \sin 100^\circ.$
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: -\sqrt{3}
extra_info.question: Simplify $\tan 100^\circ + 4 \sin 100^\circ.$
extra_info.split: test