Directed Graphs and Network Flows · Superchallenge problem

Seven teams $1,\dots,7$ play a round robin where team $i$ beats team $j$ exactly when $j-i \equiv 1, 2,$ or $3 \pmod 7$ (a rotational tournament: results depend only on how far apart the two team numbers are, cyclically). Counting two Hamiltonian cycles as the same only when they visit the teams in exactly the same cyclic order and direction, how many Hamiltonian cycles does this tournament have?