Dan and Sam play a game in which the first to start says the number \(1\), the next says \(2\), and the one who's next must say an integer number between the number previously said and its double (but not including).
For example, Dan begins saying \(1\), then Sam says \(2\), and then Dan can say whichever number he wants between \(2\) and \(4\); as the only integer between \(2\) and \(4\) is \(3\), he must say \(3\). Then, Sam can choose any number between \(3\) and \(6\); that is, he can say either \(4\) or \(5\).
The game finishes when someone reaches \(200\) (who is the winner). If Dan begins, what's the minimum quantity of numbers that the one who has a winning strategy must say, to win?
Submit \(0\) if you think that nobody has a winning strategy.
Assumptions: Both players want to win the game.