$\mathcal Z$-transform if the output is given
Since

$$Y(z) = H(z)X(z)$$ therefore
$$X(z)=\frac{Y(z)}{H(z)}$$

Now,
$$y[n]=\big(\frac{1}{3}\big)^n u[n] \longleftrightarrow Y(z)=\frac{1}{1-\frac{1}{3}z^{-1}}$$ and
$$h[n]=\big(\frac{2}{3}\big)^n u[n] + 2\big(\frac{1}{5}\big)^n u[n] \longleftrightarrow H(z)=\frac{1}{1-\frac{2}{3}z^{-1}}+\frac{2}{1-\frac{1}{5}z^{-1}}$$

will you please perform the necessary arithmetic... to find
$$X(z) = \frac{\frac{1}{1-\frac{1}{3}z^{-1}}}{\frac{1}{1-\frac{2}{3}z^{-1}}+\frac{2}{1-\frac{1}{5}z^{-1}}}$$

Lets also display the further steps:
\begin{align}
H(z) &amp;=\frac{1}{1-\frac{2}{3}z^{-1}}+\frac{2}{1-\frac{1}{5}z^{-1}} = \frac{3-\frac{23}{15}z^{-1}}{(1 - \frac{2}{3}z^{-1})(1-\frac{1}{5}z^{-1})} = \frac{3(1-\frac{23}{45}z^{-1})}{(1 - \frac{2}{3}z^{-1})(1-\frac{1}{5}z^{-1})}\\
Y(z) &amp;=\frac{1}{1-\frac{1}{3}z^{-1}}\\ \\
X(z) &amp;= Y(z)/H(z) = \frac{1}{1-\frac{1}{3}z^{-1}} \frac{(1 - \frac{2}{3}z^{-1})(1-\frac{1}{5}z^{-1})}{3(1-\frac{23}{45}z^{-1})}\\ \\
X(z) &amp;= \frac{\frac{1}{3}(1 - \frac{2}{3}z^{-1})(1-\frac{1}{5}z^{-1})}{(1-\frac{1}{3}z^{-1})(1-\frac{23}{45}z^{-1})} ~~~~~~\scriptstyle{\text{Performing partial fraction expansion on X(z), results in:}}\\ \\
X(z) &amp;= 0.26087 + \frac{0.25}{1-\frac{1}{3}z^{-1}} + \frac{-0.17754}{1-\frac{23}{45}z^{-1}} ~~~~\scriptstyle{\text{Transform each term back into time by a table...}}\\ \\
x[n] &amp;= 0.26087  \cdot \delta[n] + 0.25 \big(\frac{1}{3}\big)^n u[n] - 0.17754 \big(\frac{23}{45}\big)^n u[n] \\  
\end{align}

Sorry that the partial fraction expansion resulted in  numeric format, rather than algebraic...