A sausage factory can produce European wieners at a rate of $500$ kg per day. It supplies wieners to local stores and restaurants at a steady rate of $100$ kg per day. The cost to prepare the equipment for producing European wieners is $$12.$ Annual holding cost is $$4/$kg of wieners. The factory operates $300$ days a year. Calculate:

$*$ The optimal production run quantity.

$*$ The number of production runs per year.

$*$ The length $($in days$)$ of a production run.