The integer numbers from $1$ to $2002$ are written on a blackboard in increasing order: $1, 2, \ldots, 2001, 2002$. After that, someone erases the numbers in the $(3k+1)$-th positions, i.e., $1, 4, 7, \ldots$. Then, the same person erases the numbers in the $(3k+1)$-th positions of the new list (in this case, $2, 5, 9, \ldots$). This process is repeated until only one number remains. What is this number?

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