solution: The given expression expands to $\frac{1}{b^2} - a^2$.  Thus we want $b$ to have the least possible magnitude and $a$ to also have the least possible magnitude.  Our maximum value is therefore $\frac{1}{3^2} - (-2)^2 = \boxed{-\frac{35}{9}}$.
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  If $-6\leq a \leq -2$ and $3 \leq b \leq 5$, what is the greatest possible value of $\displaystyle\left(a+\frac{1}{b}\right)\left(\frac{1}{b}-a\right) $?  Express your answer as a common fraction.
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: -\frac{35}{9}
extra_info.question: If $-6\leq a \leq -2$ and $3 \leq b \leq 5$, what is the greatest possible value of $\displaystyle\left(a+\frac{1}{b}\right)\left(\frac{1}{b}-a\right) $?  Express your answer as a common fraction.
extra_info.split: test