GAIC Math 2024 Problem 61

For some positive integers $n$, there exists a positive integer $k\geq 2$ such that for positive integers $x_1,x_2,...,x_k$ satisfying the given condition, $\sum\limits_{i=1}^{k-1}x_ix_{i+1}=n$, $\sum\limits_{i=1}^{k}x_i=2019$ the number of such positive integers is _____.