2009 AIME I Question 14

For $t = 1,$ $2,$ $3,$ $4,$ define $S_t = \sum_{i=1}^{350} a_i^t,$ where $a_i \in \{1, 2, 3, 4\}.$ If $S_1 = 513$ and $S_4 = 4745,$ find the minimum possible value for $S_2.$