Esmeralda has created a special knight to play on quadrilateral boards that are identical to chessboards. If a knight is in a square, it can move to another square by moving 1 square in one direction and 3 squares in a perpendicular direction (which is a diagonal of a $2 \times 4$ rectangle instead of $2 \times 3$ like in chess). In this movement, it doesn't land on the squares between the beginning square and the final square it lands on.

A trip of the length $n$ of the knight is a sequence of $n$ squares $C_1, C_2, \ldots, C_n$ which are all distinct such that the knight starts at the $C_1$ square and for each $i$ from $1$ to $n-1$, it can use the movement described before to go from the $C_i$ square to the $C_{i+1}$.

Determine the greatest $N \in \mathbb{N}$ such that there exists a path of the knight with length $N$ on a $5 \times 5$ board.

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