Bounded CDF II

Let $X$ be a continuous random variable with CDF $F(x)$, PDF $f(x)$, and bounded support $[a,b]$. Let $Y$ be a random variable with CDF $G(y) = \frac{F(y)}{2-F(y)}$. Find the best possible lower bound for $\mathbb{E}\left[ Y \right]$ when $\mathbb{E}\left[ X \right] = 4$. If no such bound exists, enter $-100$.