Pythagorean Triples, $a^2+b^2=c^2$ and $b^2+c^2=d^2$. Is it possible?

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I'm trying to solve a problem, and it would be solved if I could find a solution to the above system, $a^2+b^2=c^2$ and $b^2+c^2=d^2$ for natural numbers.

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This is impossible by Fermat's right triangle theorem.