Solving polynomial systems with homotopy. Where is the bottleneck?

Question

I have a system of polynomial equations with $n$ unknowns (where $n$ can be between 3 and 20) and that is known to have at least $n!$ isolated solutions.

I want to solve this system numerically, but if I plug it in an algebraic computing system (Macaulay2 or Bertini) and try to solve it with homotopy continuation, the PC just remains blocked, and I can't find any solutions.

I am trying to understand why this happens, with the following consideration: all the $n!$ solutions are in some way equivalent (I mean that if $(x_1,..x_n)$, than the other solutions are just permutations of this array), so I am interested in computing just $1$ solution, not all the $n!$ solutions. My questions:

1. Where is the bottleneck of the homotopy method, in the number of solutions or in the number of variables?
2. The fact that I am interested only in one solution, may speed up the method just analyzing a single homotopy path?
3. Is there a way in Macaulay2 to calculated only a single solution, and not all of them?

Thanks!