For real numbers $x_1, x_2, x_3, \dots, x_{333} \in [-1, \infty)$, let $S_k = \sum_{i = 1}^{333} x_i^k$ for each $k$. If $S_2 = 777$, compute the least possible value of $S_3$.

Let's think step by step and output the final answer within \boxed{}.
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