solution: We use the property that $a \equiv b \pmod{m}$ implies $a^c \equiv b^c \pmod{m}$.
  
  Since $129 \equiv -3 \pmod{11}$ and $96 \equiv -3 \pmod{11}$, we have  $$129^{34}+96^{38} \equiv (-3)^{34}+(-3)^{38} \equiv 3^{34}+3^{38} \pmod{11}.$$Since $3^5 \equiv 1 \pmod{11},$ we can see that $3^{34} = (3^5)^{6} \cdot 3^4$ and $3^{38} = (3^5)^{7} \cdot 3^3.$
  
  Then,  \begin{align*}
  129^{34}+96^{38}&\equiv (3^5)^{6} \cdot 3^4 + (3^5)^{7} \cdot 3^3\\
  & \equiv 3^4 + 3^3\\
  & \equiv 81 + 27\\
  & \equiv 108 \\
  &\equiv \boxed{9} \pmod{11}.
  \end{align*}
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  What is the remainder when $129^{34}+96^{38}$ is divided by $11$?
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 9
extra_info.question: What is the remainder when $129^{34}+96^{38}$ is divided by $11$?
extra_info.split: test