Distance two Heegaard splittings, JSJ decompositions and ranks of 3-manifolds

For any pair of integers g and n with g ⩾ 3 and 1 ⩽ n ⩽ g, we build a 3-manifold with a distance-2, genus-g Heegaard splitting so that (1) it contains n pairwise disjoint and nonisotopic essential tori; (2) after it is cut open along these tori, one resulting piece is hyperbolic while the others are small Seifert fibered spaces; (3) it provides a substantial result to the rank versus genus problem. These generalize a result in Qiu and Zou (2019).