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Suppose that $a$ is a number such that the system of equations

$|2x| − y = 5$

$x − |2y + 2| = a$

has an odd number of distinct solutions. What is the product of all possible values of $a$?

This is a middle-school level math problem, I searched it but couldn't find a solution. Thanks for any help!

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To have an odd number of solutions, either $2x$ or $2y+2$ has to be zero.

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