  We present a computationally efficient algorithm that is suitable for graphic
processing unit implementation. This algorithm enables the identification of
all weak pseudo-manifolds that meet specific facet conditions, drawn from a
given input set. We employ this approach to enumerate toric colorable seeds.
Consequently, we achieve a comprehensive characterization of
$(n-1)$-dimensional PL spheres with $n+4$ vertices that possess a maximal
Buchstaber number.
  A primary focus of this research is the fundamental categorization of
non-singular complete toric varieties of Picard number $4$. This classification
serves as a valuable tool for addressing questions related to toric manifolds
of Picard number $4$. Notably, we have determined which of these manifolds
satisfy equality within an inequality regarding the number of minimal
components in their rational curve space. This addresses a question posed by
Chen, Fu, and Hwang in 2014 for this specific case.
