Consider the following scoring system with a slightly unusual rule. There are two players, A and B, who play $N$ games. They both start with one point each. When one of them wins, they get points until their score is coprime with the opponent’s score.
For example, if the winners are A, A, B, B, B, B, A, A, A, B, B in that order, the scores are as follows.
| Winner | A | A | B | B | B | B | A | A | A | B | B |
|---|---|---|---|---|---|---|---|---|---|---|---|
| A's score | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 5 | 6 | 6 | 6 |
| B's score | 1 | 1 | 2 | 4 | 5 | 7 | 7 | 7 | 7 | 11 | 13 |
Suppose that, after $N$ games, A and B have a final score of $a$ and $b$, where $a$ and $b$ are coprime. Are the possible values of $N$ consecutive natural numbers for any pair of $a$ and $b$?
For example, if $a = 4, b = 5$, the possible $N$ and the winners are as follows.
$N = 4$: ABAB
$N = 5$: AAABB, ABBAA, BABBA, BBAAB
$N = 6$: AABBBA, BBABAA, BBBABA
$N = 7$: BBBBAAA
I have checked up to $N< 1000$, then for any pair of $a$ and $b$, the possible $N$ were all consecutive natural numbers.