$(34$ pts$.)$ You are designing a mission to pass close to a mysterious object in a circular orbit about the sun with a radius of $2\mathrm{.}8{10}^{11}m.$ You want to incorporate a Mars flyby to gain velocity such that your spacecraft can reach that distance. You have tailored its flyby of Mars so that it will have a distance parameter, $d=8\mathrm{.}0{10}^{6}m.$ Obviously, you have designed the mission so that your spacecraft will encounter Mars' Sphere of Influence $($SOI$)$ inside of Mars' orbit $($i$.$e$.$ between Mars and the Sun$).$ Your spacecraft will leave a circular parking orbit about the Earth which has a radius of $6\mathrm{.}7{10}^{6}m.$ by applying a $V=3,743\frac{m}{sec}$ tangent to its orbit and the resulting residual velocity will be aligned with the Earth's orbital velocity. What velocity will your spacecraft gain from the Mars flyby$,$ and will you be able to reach the mysterious object? Again, you may assume that Earth and Mars are in coplanar orbits.