A sequence $a_1,a_2,\ldots$ of positive integers satisfies the following properties. $a_1 = 1$ $a_{3n+1} = 2a_n + 1$ $a_{n+1}\ge a_n$ $a_{2001} = 200$ Find the value of $a_{1000}$. Note. In the original statement of the problem, there was an extra condition: every positive integer appears at least once in the sequence. However, with this extra condition, there is no solution, i.e., no such sequence exists. (Try to prove it.) The problem as written above does have a solution.