Eccoci.

Given an odd prime $p$, put the numbers $1,...,p-1$ into $\frac{p-1}{2}$ pairs such that each number appears exactly once. Each pair $(a,b)$ has a cost of $ab \bmod p$. For example, if $p=5$ the pair $(3,4)$ has a cost of $12 \bmod 5 = 2$.

The total cost of a pairing is the sum of the costs of its pairs. We say that such pairing is optimal if its total cost is minimal for that $p$.

For example, if $p = 5$, then there is a unique optimal pairing: $(1, 2), (3, 4)$, with total cost of $2 + 2 = 4$.

The cost product of a pairing is the product of the costs of its pairs. For example, the cost product of the optimal pairing for $p = 5$ is $2 \cdot 2 = 4$.

It turns out that all optimal pairings for $p = 2\,000\,000\,011$ have the same cost product.

Find the value of this product.