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First of all, you need to acknowledge the fact that in reality there's no such thing as reversible process. It can be used as a simplification for systems in which changes are very slowly. Regarding entropy I think the intuitive way to look at entropy is as the "degree of disorder" inside a system. This definition comes from statistical thermo where entropy is related to the quantum microstates of a system (Boltzmann: S=k*lnW) In short, entropy is related to the number of ways atoms and molecules of the thermodynamic system can be arranged on energy levels. As you can imagine if you have a higher degree a disorder in the system you will have a higher number of ways the atoms and molecules are arranged on energy levels. Now coming back to your question: imagine you move very very slowly a piston inside a cylinder then you pull back the piston slowly and put it at the initial location. You can imagine that the change in the "degree of disorder" you create in the system is basically zero. That's why you can say that entropy is constant in reversible systems. This way of looking at entropy should also give you an insight on why there's no such thing as reversible process in real-life since quantum mechanics it is impossible to have the same microstates at beginning of a process and at the end of it.

I hope this intuitive approach will be helpful to you!

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