You have an $n\times n$ grid, with each cell containing a lightbulb. On a move, you may select a lightbulb, and toggle the state of that lightbulb, and all other lightbulbs sharing a row or column. For example, toggling the red $0$ results in $$\begin{bmatrix} 1 & \color{#c20}{0}& 1 \\\ 0 & 0 & 0 \\\ 1 & 1 & 0 \end{bmatrix}\longrightarrow \begin{bmatrix} 0 & 1& 0 \\\ 0 & 1 & 0 \\\ 1 & 0 & 0 \end{bmatrix} $$ where $1$ and $0$ represent lightbulb states (either lit or unlit).
Find the largest positive integer $k$, in terms of $n$, for which you can always arrive at a configuration with at least $k$ lit lightbulbs regardless of the starting configuration.