This question is rather broad so I am hoping to get references that I can look deeper into, algorithms that might help me (including theoretical exponential algorithms), or at least terms that might help me in my search. The general question can be phrased as follows:
Question: Suppose a group $G$ is generated by $G=\langle x_1,x_2,...,x_n,y_1,y_2,...,y_m\rangle$ (this is not necessarily the minimal generating set of $G$). For a given $s\in G$, what is the minimal number $k$ such that
$$s=X_0\prod_{i=1}^k\left[ y_{j_i}X_i\right]$$
(where $X_i\in \langle x_1,x_2,...,x_n\rangle$)? Note that by this formulation $k$ is allowed to be greater than $m$ since $y_i^r$ would be counted $r$ times. Stated more informally, what is the minimal number of $y_i$ that are required to generate $s$?
Background: For my specific problem, I am dealing with a group $G=\langle x_1,x_2,...,x_8,y_1,y_2,...,y_{12}\rangle$ that is otherwise difficult to describe (it is a certain subset of $16\times 16$ unitary matrices). For the particular element in question I have found ways to generate it that uses $k=10$ and $k=6$. However, I conjecture that it is possible to make said element using only $k=2$ (based off of similar yet simpler groups). If it helps (perhaps in describing some exponential algorithm?) the elements of $\langle x_1,x_2,...,x_8\rangle$ have order at most $4$ while the elements of $\langle y_1,y_2,...,y_{12}\rangle$ have order at most $2$.