A line in the plane is called sunny if it is not parallel to any of the $x$-axis, the $y$-axis, or the line $y=x$. Let $n$ be a positive integer. Determine all nonnegative integers $k$ such that there exist $n$ lines in the plane satisfying both of the following:

(i) For all positive integers $i$ and $j$ with $1\le i,j\le n$, the point $(i,j)$ is on at least one of the lines.

(ii) Exactly $k$ of the $n$ lines are sunny.

Step 1: We need $n$ lines covering all $n^2$ lattice points $(i,j)$ with $1\le i,j\le n$, with exactly $k$ being 'sunny' (not parallel to $x$-axis, $y$-axis, or $y=x$).

Step 2: Non-sunny lines are horizontal ($y=c$), vertical ($x=c$), or diagonal ($y=x+c$). $n$ horizontal lines $y=1,\dots ,n$ cover all points with $k=0$.

Step 3: To introduce sunny lines, replace some horizontal/vertical/diagonal lines with sunny lines that cover the same points. Each sunny line covers at most $n$ points from the grid. Systematic constructions show which values of $k$ are achievable.

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