Symmetry

Farmer John loves symmetry and is currently placing cows on his field, which is a rectangular grid of size n \times m.

To preserve symmetry, Farmer John proceeds as follows. First, he places a cow in the central cell of the field. If such a cell does not exist, he stops the process. Then he divides the field into four rectangular subfields of equal size, separated by the row and the column passing through the central cell containing the cow. After that, he independently applies the same procedure to each of the resulting subfields.

This process continues for smaller and smaller fields as long as the current field has a central cell and can be divided in the described manner.

For example, if n = 7 and m = 15, Farmer John will first place a cow in the cell with coordinates $(4,8)$, after which he will divide the field into four subfields of size 3 \times 7. In each such subfield, he will place a cow in the cell $(2,4)$ and then divide each of them into four subfields of size 1 \times 3. This process is illustrated below (cells containing cows are marked with the letter $C$):

A total of $21$ cows are required for such a field.

On the other hand, if n = m = 5, Farmer John will place only one cow, since the resulting 2 \times 2 subfields do not have central cells.

Help Farmer John determine how many cows in total he will need to place on the field.