Identify products $A$ and $B$ in the following reaction sequence:
$1$-methylcyclopentene $\xrightarrow{\text{dil. } KMnO_4, 273 \ K} A$ $\xrightarrow{CrO_3} B$

• A
  $A: \text{1-methylcyclopentane-1,2-diol}, B: \text{2-hydroxy-2-methylcyclopentanone}$
• B
  $A: \text{1-methylcyclopentane-1,2-diol}, B: \text{2-methylcyclopentanone}$
• C
  $A: \text{2-methylcyclopentanol}, B: \text{2-methylcyclopentanone}$
• D
  $A: \text{1-methylcyclopentene oxide}, B: \text{2-methylcyclopentanone}$