The following figure shows one highway rock slope with $12m$ in height, and the dip angel is $60.$ There is one cut$-$through inclined bedding with ${}_p=35.$ On the crest of the rock slope, there is one vertical crack with $4m$ in depth, and the water depth in the crack is $3m.$ The specific gravity of the rock mass is $27\mathrm{.}9k\frac{N}{m^3},$ the cohesion is $2\mathrm{.}5$kPa, and the frictional angle is $37.$

$(1)$ Please calculate the factor of safety $($FOS$)$ of the slope;

$(2)$ Once the vertical crack is filled with water, please recalculate the FOS;

$(3)$ Let's say the anchor cable was used to reinforce the rock slope after drainage. The installation angle of the anchor cable is ${}_T=55,$ and the pre$-$stressed force is $T=400kN,$ please evaluate the FOS of the slope under this condition;

$(4)$ How about the installation angle of the anchor cable changes from $55$ to $20?$