A Strange Conundrum

Doctor Strange is trapped in an $N$-dimensional cage. He knows that in the $i^{th}$ dimension's the cage ranges from $a_i$ to $b_i$. Using his powers, he will try to reduce the size of the cage. Specifically, he will reduce the size of the cage in the $i^{th}$ dimension to $c_i$. However, the cage is dangerous! Strange has $S$ seconds to reduce the cage. Each second, Strange can move each face of