A feedback system has an amplifier with a low-frequency open-loop gain of \(5 \times 10^{4}\) and has poles at \(10^{3} \mathrm{~Hz}, 10^{5} \mathrm{~Hz}\), and \(10^{7} \mathrm{~Hz}\).

(a) Determine the frequency \(f_{180}\) at which the phase of the loop gain is 180 degrees.

(b) Determine the feedback transfer function \(\beta\) for which the phase margin of the system is \(45^{\circ}\).

(c) Using the results of part (b), determine the low-frequency closed-loop gain.