An optimal boundedness on weak $\text{bQ}$-Fano threefolds

Let $X$ be a terminal weak $\text{bQ}$-Fano threefold. We prove that $P_{-6}(X)>0$ and $P_{-8}(X)>1$. We also prove that the anti-canonical volume has a universal lower bound $-K_X^3\ge 1\mathrm{/}330$. This lower bound is optimal.