BALTIC-WAY 2025 Problem 7

Let $n$ be an even positive integer. An $n \times n \times n$ cube is composed of $n^3$ unit cubes. A set of $n$ unit cubes that forms a $1 \times 1 \times n$ box is called a needle, which can have three orientations. Find the largest integer $K$, such that it is possible to select $3K$ needles, $K$ in each orientation, such that no two of them share a unit cube.