Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

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Equality by iteratively applying $(a,b)\rightarrow [(a+1,2b)\text{ or }(2a,b+1)]$?

I play a game starting with $2$ positive integers $a$ and $b$. At each step of the game I can double one of the integers and add $1$ to the other integer. Is there always a procedure for any ...