Let $1 = x_{1} < x_{2} < \dots < x_{k} = n$ denote the sequence of all divisors $x_{1}, x_{2}, \dots, x_{k}$ of $n$ in increasing order. Find the smallest possible value of $n$ such that
\[ n = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2}. \]

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