  Given a natural number $l$ and a weak Fano $n$-fold $X$ with
$\operatorname{dim}\overline{\varphi_{-lK_X}(X)}\geq n-1$, we study the lower
bound of the anti-canonical volume and the upper bound of the anti-canonical
stability index. The method can also be used to give similar bounds for
polarised Calabi-Yau varieties.
