jaylin has a wooden cube which is painted blue on the outside. she cuts the cube into $1000$ identical cubes, some of which have some sides painted blue, then rolls the resulting cubes like dice. the probability that no blue faces land up after jaylin rolls the $1000$ cubes can be expressed as $2^a \times 3^b \times 5^c$ where $a$, $b$ and $c$ are integers. what is the value of $a b c$?