2021 April MIMC Problems/Problem 18

What can be a description of the set of solutions for this: $x^{2}+y^{2}=|2x+|2y||$ ?

$\textbf{(A)}$ Two overlapping circles with each area $2\pi$ .

$\textbf{(B)}$ Four not overlapping circles with each area $4\pi$ .

$\textbf{(C)}$ There are two overlapping circles on the right of the $y$ -axis with each area $2\pi$ and the intersection area of two overlapping circles on the left of the $y$ -axis with each area $2\pi$ .

$\textbf{(D)}$ Four overlapping circles with each area $4\pi$ .

$\textbf{(E)}$ There are two overlapping circles on the right of the $y$ -axis with each area $4\pi$ and the intersection area of two overlapping circles on the left of the $y$ -axis with each area $4\pi$ .

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2021 April MIMC Problems/Problem 18