An AISI $1020$ cold$-$drawn steel shaft with the geometry shown in the figure carries a transverse load of $7$ kN and a torque of $107$ N $$ m$.$ Examine the shaft for strength and deflection. If the largest allowable slope at the bearings is $0\mathrm{.}001$ rad and at the gear mesh is $0\mathrm{.}0005$ rad, what is the factor of safety guarding against damaging distortion? Using the DE$-$Gerber criterion, what is the factor of safety guarding against a fatigue failure? If the shaft turns out to be unsatisfactory, what would you recommend to correct the problem?