Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous odd function, vanishing exactly at one point and satisfying $f(1) = \tfrac{1}{2}$. If $\lim_{x\to 1} \frac{F(x)}{G(x)} = \frac{1}{14},$ where $F(x) = \int_{-1}^{x} f(t)\,dt, \qquad G(x) = \int_{-1}^{x} t\,|f(f(t))|\,dt,$ then the value of $f\!\left(\tfrac{1}{2}\right)$ is