$4.$ A total of $200$ kg dry organic refuse is collected and the chemical composition is C$8$H$18$O$6$N$2.$

$1)$ Write and balance the chemical reaction equation when the refuse is added to an anaerobic digestor for methane production.

$2)$ Estimate i$)$ the volume of methane gas produced at standard temperature and pressure and ii$)$ the energy recovered in terms of methane production through anaerobic digestion? Assume all the $200$ kg organic refuse is decomposed and the energy density of methane is about $50$ MJ$/$kg$.$