9.76 On a sufficiently large chessboard composed of squares with an area of 1, a non-self-intersecting closed broken line is drawn, with each edge of the broken line lying on the grid lines. It is known that there are $k$ black squares inside the broken line. Find the maximum area enclosed by the broken line.

保留源文本的换行和格式，翻译结果如下：

9.76 On a sufficiently large chessboard composed of squares with an area of 1, a non-self-intersecting closed broken line is drawn, with each edge of the broken line lying on the grid lines. It is known that there are $k$ black squares inside the broken line. Find the maximum area enclosed by the broken line.
[Solution] On a chessboard, each black square is adjacent to 4 white squares, so $k$ black squares have a total of $4k$ white neighboring squares. On the other hand, since the polyline does not intersect itself and each side lies on a grid line, we can use a polyline with each side parallel to the grid lines to connect the centers of the $k$ black squares. Clearly, this polyline alternates between black and white squares. Therefore, there are at least $k-1$ white squares between the $k$ black squares, and each white square is adjacent to exactly two of the $k$ black squares. It is evident that in the process of counting neighboring squares, at least $k-1$ white squares are counted twice. Thus, the area enclosed by the polyline does not exceed $4k+1$.

Obviously, when the $k$ black squares are in a single row, the maximum area enclosed is exactly $4k+1$.