$[8$ pts$]$ An object is moving along a scaled line. At time $t$ seconds, its acceleration if given by $a(t)=e^t+6t$ meters per second squared. At time $t=0$ seconds, its position is $s(0)=7$ meters, and its velocity is $v(0)=3$ meters per second. Find the object's position $s(t)$ at time $t$ seconds.