What is the parity of $$$s$$$ after an operation?

After the first operation, $$$s$$$ is always odd. You can only earn points when $$$a_i$$$ is odd after this operation. However, during the first operation, you can only earn a point if $$$a_i$$$ is even.

Let $$$c_0$$$ and $$$c_1$$$ be the number of even and odd numbers, respectively.

If there is at least one even number, you can place it first, making the answer $$$c_1 + 1$$$. Otherwise, since no points can be earned during the first operation, the answer is $$$c_1 - 1$$$.

Time complexity: $$$O(n)$$$.