solution: Isolating $\tan x^\circ,$ we find
  \begin{align*}
  \tan x &= \frac{\tan 53^\circ + \tan 81^\circ}{\tan 53^\circ \tan 81^\circ - 1} \\
  &= -\frac{\tan 53^\circ + \tan 81^\circ}{1 - \tan 53^\circ \tan 81^\circ}.
  \end{align*}From the angle addition formula, this is equal to
  \[-\tan (53^\circ + 81^\circ) = -\tan 134^\circ = \tan 46^\circ.\]Therefore, $x = \boxed{46}.$
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  For some value of $x,$ $0 < x < 180,$
  \[\tan 53^\circ \tan 81^\circ \tan x^\circ = \tan 53^\circ + \tan 81^\circ + \tan x^\circ.\]Find $x.$
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 46
extra_info.question: For some value of $x,$ $0 < x < 180,$
  \[\tan 53^\circ \tan 81^\circ \tan x^\circ = \tan 53^\circ + \tan 81^\circ + \tan x^\circ.\]Find $x.$
extra_info.split: test