Given $n$ and $t$ lengths  $ l_i, 1\leq l_1<l_2<\cdots<l_t\leq n-1$, of directed diagonals within an $n$-gon such that $l_1+\cdots+l_t\neq 0$. Does it exist a directed path within the n-gon using each of the given lengths exactly once (and no vertex twice)?

Here the length of a directed diagonal within any kind of an n-gon is the number of sides of the n-gon between the starting and the end vertex counted in a fixed direction. A directed path is a sequence $(d_1,d_2,\ldots,d_t)$ of directed diagonals such that the end vertex of $d_i$ is the starting vertex of $d_{i+1}$ for $i=1,2,…,t-1$.

In number theory, it is equivalent to: For any subset of $\{1,2,…,n-1\}$ with the sum of its elements $\neq 0 (mod n)$ is there any a permutation of the elements of this subset such that no set of consecutive elements in this permutation has sum $=0 (mod n)$.

I have a proof for case $t\leq 6$. However, I wonder is it true or not in general.
Many thanks