solution: The area of the inside circle (region $X$) is $\pi\cdot 4^2=16\pi.$
  
  Using a similar technique, the area of the middle ring (region $Y$) is $$\pi\cdot 6^2-\pi\cdot 4^2=36\pi-16\pi = 20\pi.$$ Also, the area of the outer ring (region $Z$) is $$\pi\cdot 7^2-\pi\cdot 6^2=49\pi - 36\pi = 13\pi.$$ Therefore, region $Y$ has the largest area and region $Z$ has the smallest area. The difference in their areas is $20\pi-13\pi = \boxed{7\pi}.$
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  In the diagram, the three concentric circles have radii of $4,$ $6,$ and $7.$ Three regions are labeled $X,$ $Y,$ or $Z$ below. Of these three regions, what is the difference between the area of the region with the greatest area and the area of the region with the smallest area? Express your answer in exact form.
  
  [asy]
  import graph;
  filldraw(circle((0,0),7), lightgray, black+linewidth(1));
  filldraw(circle((0,0),6), gray, black+linewidth(1));
  filldraw(circle((0,0),4), white, black+linewidth(1));
  dot((0,0));
  label("$X$",(2,0));
  label("$Y$",(5,0));
  label("$Z$",(6.5,0));
  [/asy]
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 7\pi
extra_info.question: In the diagram, the three concentric circles have radii of $4,$ $6,$ and $7.$ Three regions are labeled $X,$ $Y,$ or $Z$ below. Of these three regions, what is the difference between the area of the region with the greatest area and the area of the region with the smallest area? Express your answer in exact form.
  
  [asy]
  import graph;
  filldraw(circle((0,0),7), lightgray, black+linewidth(1));
  filldraw(circle((0,0),6), gray, black+linewidth(1));
  filldraw(circle((0,0),4), white, black+linewidth(1));
  dot((0,0));
  label("$X$",(2,0));
  label("$Y$",(5,0));
  label("$Z$",(6.5,0));
  [/asy]
extra_info.split: test