Limit of a Sequence II

Let $(x_n)$ be a sequence defined by $x_1 = 2026$ and $$x_{n+1} = x_n + \frac{1}{x_n}, \qquad \forall n \ge 1.$$ Let $S$ be the set of all positive real numbers $C$ such that $$x_n < \sqrt{2n + C} \qquad \text{for all } n \in \mathbb{N}^*.$$ Find the sum of all elements of $S$.