The ship *Meridiano do Bacalhau* conducts its fishing business over a period of $64$ days. Each day, the captain chooses a direction, either north or south, and the ship sails in that direction. On the first day, the ship sails $1$ mile; on the second day, it sails $2$ miles; generally, on the $n$-th day, it sails $n$ miles. After the $64$-th day, the ship is $2014$ miles north of its initial position. What is the greatest number of days that the ship could have sailed south?

Let's think step by step and output the final answer within \boxed{}.
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