solution: In cylindrical coordinates, $z$ simply denotes the $z$-coordinate of a point.  Thus, for a fixed $z$-coordinate $c,$ all the points lie on a plane that is parallel to the $xy$-plane.  The answer is $\boxed{\text{(C)}}.$
  
  [asy]
  import three;
  import solids;
  
  size(200);
  currentprojection = perspective(6,3,2);
  currentlight = (1,0,1);
  real theta = 120;
  
  draw((-2,0,0)--(2,0,0));
  draw((0,-2,0)--(0,2,0));
  draw(surface((1,1,0.5)--(1,-1,0.5)--(-1,-1,0.5)--(-1,1,0.5)--cycle),gray(0.99));
  draw((0,0,-2)--(0,0,0.2));
  draw((0,0,0.5)--(0,0,2));
  
  label("$x$", (2,0,0), SW);
  label("$y$", (0,2,0), E);
  label("$z$", (0,0,2), N);
  label("$z = c$", (-1,1,0.5), E);
  [/asy]
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  For a constant $c,$ in cylindrical coordinates $(r,\theta,z),$ find the shape described by the equation
  \[z = c.\](A) Line
  (B) Circle
  (C) Plane
  (D) Sphere
  (E) Cylinder
  (F) Cone
  
  Enter the letter of the correct option.
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: \text{(C)}
extra_info.question: For a constant $c,$ in cylindrical coordinates $(r,\theta,z),$ find the shape described by the equation
  \[z = c.\](A) Line
  (B) Circle
  (C) Plane
  (D) Sphere
  (E) Cylinder
  (F) Cone
  
  Enter the letter of the correct option.
extra_info.split: test