  We prove a version of Jonsson-Musta\c{t}\v{a}'s Conjecture, which says for
any graded sequence of ideals, there exists a quasi-monomial valuation
computing its log canonical threshold. As a corollary, we confirm Chi Li's
conjecture that a minimizer of the normalized volume function is always
quasi-monomial.
  Applying our techniques to a family of klt singularities, we show that the
volume of klt singularities is a constructible function. As a corollary, we
prove that in a family of klt log Fano pairs, the K-semistable ones form a
Zariski open set. Together with [Jia17], we conclude that all K-semistable klt
Fano varieties with a fixed dimension and volume are parametrized by an Artin
stack of finite type, which then admits a separated good moduli space by [BX18,
ABHLX19], whose geometric points parametrize K-polystable klt Fano varieties.
