A $5 \times 5$ grid of squares is filled with integers. Define a rectangle as "corner-odd" if its sides are grid lines and the sum of the integers at its four corners is an odd number. What is the maximum possible number of corner-odd rectangles within the grid?

Note: A rectangle must have four distinct corners to be considered "corner-odd"; i.e., no $1 \times k$ rectangle can be "corner-odd" for any positive integer $k$.

Let's think step by step and output the final answer within \boxed{}.
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