OME 2022 Problem 2

Let $a,b,c,d$ be four positive real numbers. If they satisfy $$a+b+\frac{1}{ab}=c+d+\frac{1}{cd} \text{and} \frac1a+\frac1b+ab=\frac1c+\frac1d+cd$$ then prove that at least two of the values $a,b,c,d$ are equal.