A mixture of benzene and toluene is fed to an intermediate point of a continuous distillation column. The feed contains $45$mol$\%$ benzene and $55$mol$\%$ toluene. A distillate containing $92$ mol$\%$ benzene and $8$mol$\%$ toluene is obtained. The bottom product contains $15$mol$\%$ benzene and $85$mol$\%$ toluene. One mole of distillate product is removed for every $4$mol returned to the unit as reflux. The feed is a sub$-$cooled liquid and the value of q$-$factor is $1\mathrm{.}204.$ The column operates at atmospheric pressure. The equilibrium data is given below:

$\backslash$table$[[x,0,0\mathrm{.}1,0\mathrm{.}2,0\mathrm{.}3,0\mathrm{.}4,0\mathrm{.}5,0\mathrm{.}6,0\mathrm{.}7,0\mathrm{.}8,0\mathrm{.}9,1],[y,0,0\mathrm{.}22,0\mathrm{.}38,0\mathrm{.}51,0\mathrm{.}63,0\mathrm{.}7,0\mathrm{.}78,0\mathrm{.}85,0\mathrm{.}91,0\mathrm{.}96,1]]$

Using the McCabe$-$Thiele graphical method, answer to the following questions:

$($i$)$ How many theoretical stages are required to achieve the given separation? $($ii$)$ What is the value of reflux ratio? $($iii$)$ What is the feed tray location?