probability - Binomial random variable expectation bound

Let $X,Y \sim \text{Bin}(n,0.5)$ for some positive $n$.

What is a lower bound for $\mathbb{E}(XY)$? When is it achieved?

My try:

I got confused by the following two results and couldn't proceed!

By Jensen's, $\mathbb{E}(XY) \geq \mathbb{E}(X)\mathbb{E}(Y)$. But we also know that $\text{cov}(X,Y)=\mathbb{E}(XY) - \mathbb{E}(X)\mathbb{E}(Y)$, and this quantity can be negative!

Please help me to proceed. Thanks in advance!