How to solve simultaneous logical equations?

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Suppose I have two equations $$y = (x\oplus a)+b \\ x = (y\oplus c)+d$$ All numbers are in base 10.

$\oplus$ is the XOR operation for binary numbers.

How do I solve such equations manually, or is there some Mathematica function I could use? I prefer not using matrices.

I did try using a C++ program to brute force it within a range, but it is not an ideal solution.