"To pursue this avenue" means to continue exploring or investigating a specific approach or option to solve a problem or achieve a goal. Full definition
I have grown professionally and my skills have developed since then, but I will always have the flexibility to return to the clinical setting, or even pursue another avenue of nursing. (careertoolkit.com)
Scientists have a duty to remain unbiased and to aggressively pursue all avenues until a general consensus is arrived at. (illconsidered.blogspot.ru)
We will pursue every avenue available and will not rest until we have obtained justice for you. (cochranfirm.com)