Eventual Pairs

Logic Level 5

\[ (10,4) \rightarrow (20,5) \rightarrow (21,10) \rightarrow (42,11) \rightarrow (43,22) \rightarrow (44,44) \]

I have two integers \(A\) and \(B\). Each turn, I must double one number and add 1 to the other. I repeat this process, with the goal of making the integers equal to each other. The above is an example of how we can start from \( (10,4) \) and get to two equal integers.

Given that I have the initial pairs of integers \(A = 300,B=301 \) and that after \(M\) steps I have the pairs \((N,N) \) for some integer \(N\). What is the value of \(N\) such that \(M\) is minimized?

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