ABC is a toy manufacturer. It manufactures both toy components and finished toys. For a specific type of toys, the manufacturer can sell the components to an assembly company for $$12$ per set, or assemble the toys itself. The finished toys can be sold for $$17$ each. ABC has a total capacity of $500,000$ hours, which could be used for both producing toy components and assembling. One capacity hour can be used to produce $30$ sets of toy components or to assemble $15$ units of toys. All the related costs for the manufacturing and assembling are listed below. No other costs and tax are considered. The manufacturing overhead cost is fixed and allocated to the units produced based on their production time. We assume that the demand is high enough so that the manufacturing capacity is always utilized fully.

The direct materials, direct labor and manufacturing overhead for each set of toy components are $$4\mathrm{.}60,$ $$3\mathrm{.}00$ and $$1\mathrm{.}00,$ respectively.

The additional direct materials, direct labor and manufacturing overhead for each unit of assembly are $$1\mathrm{.}20,$ $$0\mathrm{.}50$ and $$2\mathrm{.}00,$ respectively.

$(1)$ The ABC executives decide to assemble the toys themselves instead of selling the components. With the expected sale price of $$17$ per unit, is this a good decision?

$(2)$ What is the lowest price for the completely assembled $($finished$)$ toys that will be acceptable for the executives$$ decision?

$(3)$ The manufacturing overhead cost in $(1)$ and $(2)$ is a fixed cost allocated to each unit. Suppose that $50\%$ of that fixed overhead cost is variable. Please reevaluate your answers in $(1)$ and $(2).$