# The First Anniversary Problem

Algebra Level 5

Let $$n$$ be a positive integer and $$x_1, x_2, \ldots, x_n$$ be integers satisfying the following conditions:

1. $$-1 \le x_i \le 2$$ for all $$i = 1, 2, \ldots, n$$
2. $$\displaystyle \sum_{i=1}^n x_i = 19$$
3. $$\displaystyle \sum_{i=1}^n x_i^2 = 99$$

Out of all possible $$n, x_1, x_2, \ldots, x_n$$ satisfying the above conditions, find the sum of the minimum and the maximum values of $$\displaystyle \sum_{i=1}^n x_i^3$$.

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