Each of 8 bricks (right rectangular prisms) has dimensions $a\times b\times c$, where $a,b,$ and $c$ are pairwise relatively prime positive integers. These bricks are arranged to form a $2\times 2\times 2$ block with overall dimensions $2a\times 2b\times 2c$. An additional brick with the same dimensions is introduced, and these bricks are reconfigured into a $1\times 3\times 3$ block with overall dimensions $a\times 3b\times 3c$. The new block is 1 unit taller, 1 unit wider, and 1 unit deeper than the old one. What is $a+b+c$?