In the USA and Canada the practice of having "house cats", which are never allowed outside, is surprisingly common. However, in most other countries of the world it is often regarded as unnatural and restricting a cat's natural desire to explore, climb trees, etc.
Personally, I think that you should allow your cat to have the freedom to explore. Regarding going in/out of the house, the best solution is to get a cat flap. They can be easily fitted into a wooden door by anyone with basic woodwork skills. If you search on Amazon for "cat flap", you will find many examples. That means the cat will be free to come in/go out whenever s/he wishes. It also means that you won't need to change the litter in a cat tray (although it is still wise to have one for emergencies).
Every best wish to you!
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