In a blackboard, the number 2012 is initially written. On each turn, the student can choose on of the three numbers: 6, 1006, 1509, and multiply it with the existing number on the board. The existing number is then erased and replaced with the new product.
The teacher has a specific integer $k > 1$ in mind, and the goal for the student is to get the number of the board to have form $n^k$ where $n$ is an integer.
Determine all values of $k$ such that, after a finite number of turns, the number of the board will have the desired form.