I am learning Z trasnform.The region of convergence is extremely important for a Z-transform.The region of convergence is extremely important for a Z-transform,but i encouter a problem that when i calculate Z-transform of convolution of two sequences ，i have no idea how to determine its region of convergence. The Z-transform of this sequnces (as belows) has three poles , and I haven't expanded this sequence into series form, and I can't use methods such as ratio discrimination to discriminate series convergence. Should I expand this sequence into series form to use methods in convergence of series？ One step further ,there is any general way to solve Z-transform's region of convergence?
$$y(n)=x_1(n+3)*x_2(-n+1)$$ $$x_1(n)=(\frac{1}{2})^{n}*u(n)$$ $$x_2(n)=(\frac{1}{3})^{n}*u(n)$$