A battery with e m f $=8\mathrm{.}30$ V and no internal resistance supplies current to the circuit shown in the figure below. When the double$-$throw switch S is open as shown in the figure, the current in the battery is $1\mathrm{.}00$ mA$.$ When the switch is closed in position a$,$ the current in the battery is $1\mathrm{.}14$ mA$.$ When the switch is closed in position b$,$ the current in the battery is $1\mathrm{.}90$ mA$.$

A circuit consists of three square loops arranged such that one larger loop is on the left and two smaller loops, one on top of the other, are on the right. The top right loop shares its left side with the top half of the left loop's right side and the bottom right loop shares its left side with the bottom half of the left loop's right side.

A battery with emf $$ is on the left side of the left loop with the positive terminal below the negative terminal. A resistor R$1$ is on the top side of the left loop. A resistor R$2$ is on the top half of the right side of the left loop and the wire terminates at point a just below the resistor. Below point a$,$ there is a gap before the wire begins again at point b$,$ which is on the bottom half of the right side of the left loop.

A resistor R$2$ is on the top side of the top right loop. The left side of the top right loop is shared with the top half of the left loop that contains a resistor R$2$ and point a just below the resistor. The bottom side of the top right loop contains a switch S that can connect either to point a$,$ point b or neither, but not both at once. It is connected to neither point a nor point b$.$

A resistor R$3$ is on the right side of the bottom right loop. The left side of the bottom right loop is shared with the bottom half of the left loop that contains point b$.$ The top side of the bottom right loop is shared with the bottom side of the top right loop that contains the switch S$.$

$($a$)$ Find the resistance R$1.$

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$($b$)$ Find the resistance R$2.$

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$($c$)$ Find the resistance R$3.$

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