2012 AMC 10B Problems/Problem 20

$20.$ Bernardo and Silvia play the following game. An integer between $0$ and $999$ , inclusive, is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds $50$ to it and passes the result to Bernardo. The winner is the last person who produces a number less than $1000$ . Let $N$ be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of $N$ ?

$(A)$ $7$ $(B)$ $8$ $(C)$ $9$ $(D)$ $10$ $(E)$ $11$

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2012 AMC 10B Problems/Problem 20