Reach 200

Dan and Sam play a game. Dan starts and says the integer $1$ , then Sam says $2$ . For each subsequent turn, the one who's next must say an integer that is strictly between the integer previously said and its double.

For example, Dan begins saying $1$ , then Sam says $2$ , and then Dan can say whichever integer 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 integer between $3$ and $6$ ; that is, he can say either $4$ or $5$ .

The person who first says $200$ is the winner. For the person who has a winning strategy, what is the minimum number of integers that he says?

Submit $0$ if you think that nobody has a winning strategy.

Assumptions: Both players want to win the game.

This is the fourth problem of the set Winning Strategies .