This is based on the well-known problem of ordering five coins with $7$ (binary) weighings. There are $120$ permutations and $128$ possible sequences of seven answers, and it is indeed possible to find a strategy which always works.

In our problem, there are $720$ possible answers, and with at most $6$ ternary weighings, theoretically we could obtain $3^6=729$ possible answers. The gap is very small, but it turns out that it is again possible to find a strategy which uses just $6$ weighings.

The easiest solution here is to generate a memorization function which, for each possible subset of possible permutations, tries all the possible questions, and returns the best one. It is possible to bound the branching, as follows:

• Always take care that there are at most $243$ consistent results after $1$ weighing, at most $81$ consistent results after $2$ weighings, etc.
• If, for the given subset of power $P$, we have already found a solution which uses $N$ weighings, and $P>3^{N-1}$, then do not look further (we have already found the best possible answer).

With these optimizations, we could write a program which generates the strategy tree, in the form of a program which could be submitted for judging. With the optimizations above, it is also possible to generate it on the fly.