PAMO 2024 Problem 3

Given an integer $n \geq 1$, Jo-Ané alternately writes crosses ($\mathcal{X}$) and circles ($\mathcal{O}$) in the cells of a square grid with $2n + 1$ rows and $2n + 1$ columns: she first writes a cross in a cell, then a circle in a second cell, then a cross in a third cell, and so on. When the table is completely filled, her score is calculated as the sum $\mathcal{X}+ \mathcal{O}$, where $\mathcal{X}$ is the number of rows containing more crosses than circles and $\mathcal{O}$ is the number of columns containing more circles than crosses.

Determine, in terms of $n$, the highest possible score that Jo-Ané can obtain..