Translation number with respect to freely generating set equals length of cyclically reduced word

Statement
Suppose $$F$$ is a finitely generated free group with freely generating set $$x_1,x_2,\dots,x_n$$. For any conjugacy class in $$F$$, the translation number for that conjugacy class with respect to the freely generating set is equal to the length of any cyclically reduced word that gives an element of the conjugacy class. Here, a cyclically reduced word is a word such that every cyclic permutation of it gives a reduced word.

For instance, the word $$x_1x_2x_1x_2x_1^{-1}x_3x_1^{-1}$$ is a reduced word but is not cyclically reduced, whereas the word $$x_2x_1x_2x_1^{-1}x_3$$, which describes an element of the same conjugacy class, is cyclically reduced.

Corollaries

 * Finitely generated free implies strongly translation-discrete