JMO 2025 Problem 3

Let $n$ be a positive integer. There exist $n$ ordered triples $$(x_1, y_1, z_1), (x_2, y_2, z_2), \dots, (x_n, y_n, z_n)$$ where each coordinate is an integer between $1$ and $100$ (inclusive), satisfying the following condition:

[center] For every infinite sequence $(a_1, a_2, a_3, \dots)$ of integers between $1$ and $100$ , there exist a positive integer $i$ and an index $j$ (with $1 \leqslant j \leqslant n$ ) such that $(a_i, a_{i+1}, a_{i+2}) = (x_j, y_j, z_j)$ . [/center]

Determine the minimum possible value of $n$ .