Fundamental group of smooth part of log fano varieties

Question

I'm interested in fundamental group of smooth part of log fano varieties.

Question.1 Is there an example of a non-Gorenstein terminal Fano 3-fold whose smooth part is simply connected?

Actually, I'm interested in classification of terminal fano 3-folds which are not necessarily Gorenstein. I think it's optimistic to expect that every terminal Fano 3-folds can be written as an quotient of terminal Gorenstein Fano 3-folds. So I want to ask;

Question 2 Is there an example of a terminal Fano 3-fold which cannot be written as an quotient of terminal Gorenstein Fano 3-folds by finite group action?

If you know an example of such Q-Fano 3-fold of either type, please let me know about it.

Answer 1

Example for Q1. Consider $\mathbb CP^3/\mathbb Z_2$ with the action $(x:y:z:t)\to (x:y:z:-t)$. The quotient has one terminal non-Goernstein singularity $(0:0:0:1)$ and the complement to this singularity is isomorphic to a line bundle ($O(2)$) over $\mathbb CP^2$, so it has $\pi_1=0$.