Prove that there exist infinitely many triples of positive integers $(d_1, d_2, d_3)$ satisfying \(3 \le d_1 \lt d_2 \lt d_3\) for which the following system admits a solution in non-negative integers $(x_{i,j,k})$:

\[ d_i d_j = \delta_{i,j} + \sum_{k=1}^{3} d_k x_{i,j,k}, \qquad \text{for all } i,j \in \{1,2,3\}, \]

where the unknowns $(x_{i,j,k})$ are completely symmetric in their indices.

Prove that there exist infinitely many triples of positive integers $(d_1, d_2, d_3)$ satisfying $3 \le d_1 < d_2 < d_3$ for which the following system admits a solution in non-negative integers $(x_{i,j,k})$:

$$d_i d_j = \delta_{i,j} + \sum_{k=1}^{3} d_k \, x_{i,j,k}, \quad \text{for all } i,j \in \{1,2,3\},$$

where the unknowns $(x_{i,j,k})$ are completely symmetric in their indices.