solution: The center of the original circle is $(0,0).$  The reflection of the point $(0,0)$ in the point $(4,1)$ is $(8,2),$ so the equation of the new circle is
  \[(x - 8)^2 + (y - 2)^2 = 25.\]This simplifies to $x^2 + y^2 - 16x - 4y + 43 = 0.$  Hence, $(a,b,c,d) = \boxed{(1,-16,-4,43)}.$
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  If each point of the circle $x^2 + y^2 = 25$ is reflected in the point $(4,1),$ the set of image points satisfies the equation
  \[x^2 + ay^2 + bx + cy + d = 0.\]Compute the ordered quadruple $(a,b,c,d)$ of real numbers.
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: (1,-16,-4,43)
extra_info.question: If each point of the circle $x^2 + y^2 = 25$ is reflected in the point $(4,1),$ the set of image points satisfies the equation
  \[x^2 + ay^2 + bx + cy + d = 0.\]Compute the ordered quadruple $(a,b,c,d)$ of real numbers.
extra_info.split: test