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If $x[n]=(\frac{1}{3})^{|n|}-(\frac{1}{2})^{|n|}u[n]$, then the region of convergence (ROC) of its Z-transform in the Z-plane will be

1. $\frac{1}{3}\lt |z|\lt 3$
2. $\frac{1}{3}\lt |z|\lt \frac{1}{2}$
3. $\frac{1}{2}\lt |z|\lt 3$
4. $\frac{1}{3}\lt |z|$