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 $\displaystyle \lim_{x \to 1} \frac{F(x)}{G(x)}=\frac{1}{14}$