JEE-Advanced 2015 (27/40)

Let $f:\mathbb R \to \mathbb R$ be a continuous odd function, which vanishes exactly at one point and $f(1)=\frac{1}{2}$ . Suppose that $F(x)=\displaystyle \int_{-1}^x f(t)dt$ for all $x \in [-1,2]$ and $G(x)=\displaystyle \int_{-1}^x t|f(f(t))|dt$ for all $x \in [-1,2]$ . Find the value of $f \left(\frac{1}{2} \right)$ if $\displaystyle \lim_{x \to 1} \frac{F(x)}{G(x)}=\frac{1}{14}$