$)$ Consider the limacon r $=24$ sin$($$)$ for $0$ $2$$,($a full period, as plotted.$)$

$(4)$i$)$ It is intuitive that there are $4$ vertical tangent lines. Locate them.

$(3)$ii$)$ The plot suggests that there is a self intersection at the pole $$ locate the values at which this occurs, and find the slope of the tangent line at respective for a parametrization $($$)=($f $($$)$ cos$($$),$ f $($$)$ sin$($$))$ with f $($$)=24$sin$($$),0$ $2$$.$

$(4)$iii$)$ Calculate the area of the region between the two $$loops$.$