# Reach 200

Logic Level 3

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.
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