Given the following sequence:

$f[n]=2^{n}u[n].$

$($a$)(16$ points$)$ Determine whether $f[n]$ is absolutely summable.

$($b$)(16$ points$)$ Determine whether the Fourier transform of $f[n]$ converges and why.

$($c$)(20$ points$)$ Let $y[n]=a^{-1}f[n],$ for what values of a does the Fourier transform of $y[n]$ converge?

$($d$)(16$ points$)$ By the definition of z$-$transform, obtain the z$-$transform $F(z)$ of $f[n]$ and its region of convergence $($ROC$).$

$($e$)(12$ points$)$ What is the difference between the regions of convergence in $($c$)$ and $($d$)?$ Comment elaborately.

$($f$)(20$ points$)$ Given that $F(z)$ where $z=3e^j,$ is the Fourier transform of a sequence $f_1[n],$ then obtain $f_1[n].$

Explain part c step by step