Directed Graphs and Network Flows · Superchallenge problem

Six teams $1,\dots,6$ play a round robin where team $i$ beats team $j$ for every pair $i<j$, EXCEPT that team $6$ beats team $1$, and team $5$ beats team $2$ (two upsets). A Hamiltonian path visits every team exactly once, always moving winner-to-loser. How many different Hamiltonian paths does this tournament have?