infinite chessboard average!

The \(n^2\) squares of an \(n \times n\) chessboard are filled with positive integers in such a way that each integer is the average of the integers on the neighbouring squares (sharing common edge and vertex). Let \(k\) be the number of all such distinct configurations. What is the value of \(\displaystyle \lim_{n \to \infty} \frac{k}{n}\)?

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