Word length in symmetrized presentations of Thompson's group $F$

Thompson's groups $F, T$ and $Z$ were introduced by Richard Thompson in the 1960's in connection with questions in logic. They have since found applications in many areas of mathematics including algebra, logic and topology, and their metric properties with respect to standard generating sets have been studied heavily. In this paper, we introduce a new family of generating sets for $F$, which we denote as $Z_n$, establish a formula for the word metric with respect to $Z_1$ and prove that $F$ has dead ends of depth at least $2$ with respect to $Z_1$.