51st Ukrainian National Mathematical Olympiad, 3rd Round

There are $2012$ piles of stones. The first pile contains $2^0$ stones, the second pile contains $2^1$ stones, the third pile contains $2^2$ stones, and so on. The $2012$-th pile contains $2^{2011}$ stones. At one step, one can pick three piles and add $2$ stones to the first pile, $3$ stones to the second pile, and $4$ stones to the third pile, or pick any three piles and take $2$ stones out of the first pile, $3$ stones out of the second pile, and $4$ stones out of the third pile, provided that each pile has enough stones. Is it possible after a finite number of such operations to get exactly $3^{1005}$ stones in each pile?