There are 2 strictly increasing functions f and g that map positive integers to positive integers. It is known that f(f(f(x)))=g(g(x)) for all positive integers x.

It is given that g(x)=ax+b.

Jasmine and Luke and perfect logicians and functional experts.

Jasmine is told the value of a+b. Luke is given a.

Jasmine: If you tell me a, I would know infinitely many values of f(x).
Luke: If I told you a, and you were given a random positive integer y, could you determine f(y).
Jasmine: It depends.
Luke: Then, if I were given any positive integer y, I could determine f(y).
Jasmine: Then so could I.

Jasmine is then given y=2019.

What value of f(2019) does she give?

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