A chemical manufacturer is setting up capacity in Europe and North America for the next $3$ years. Annual demand in each market is $2$ million kilograms $($kg$)$ and is likely to stay at that level. The two choices under consideration are building $4$ million units of capacity in North America and building $2$ million units of capacity in each of the two locations. Building two plants will incur an additional one$-$time cost of $$2$ million. The variable cost of production in North America $($for either a large or a small plant$)$ is currently $$10/$kg$,$ while the cost in Europe is $$9/$kg$.$ The current exchange rate is $1$ euro for US$$1\mathrm{.}33.$ Over each of the next three years, the dollar is expected to strengthen by $10\%$ with a probability of $0\mathrm{.}5$ or weaken by $5\%$ with a probability of $0\mathrm{.}5.$ Assume a discount factor of $10\%.$

$1.$ What is the NPV value associated with building a $4$ million units of capacity in North America?

$2.$ What is the NPV value associated with building a building $2$ million units of capacity in each of the two locations $($North America and Europe$)?$