Task $1(30$ pts $)$: Please perform a circuit analysis of the following BSF circuit that you just designed to include the following:

$(1)$ Show a detailed hand calculation of circuit analysis at low and high frequencies. Recall that at low frequencies, capacitors behave like opens; at high frequencies, capacitors behave like shorts.

$(2)$ Your hand calculation needs to show that at low frequencies, the BSF gain is about $-8\mathrm{.}3$ dB ; at high frequencies, the BSF gain is about $0$ dB $.$ Draw the $2$ circuit diagrams and analyze them.

$(3)$ Suggest a design modification that will let your modified BSF circuit produce a true band$-$stop Bode plot, with both the low and high frequency gains at $0$ dB $,$ as shown in the Bode plot below. You may modify the BSF circuit by adding or removing elements, and$/$or changing element values. Use as few circuit elements as possible.