It takes $15$ s to raise a $1,200-$kg car and the supporting $300-$kg hydraulic car$-$lift platform to a height of $2\mathrm{.}8$ m$.$ The velocity of the lift increases uniformly from zero to its maximum value at mid$-$height in $7\mathrm{.}5$ s and then decreases uniformly to zero in $7\mathrm{.}5$ s$.$ Knowing that the peak power output of the hydraulic pump is $7$ kW when the velocity is maximum, determine the maximum lift force provided by the pump.