Problem $1.(20$ points$)$ Let the market demand for sparkling water be given by $\backslash ($ Q$=\backslash )\backslash (500-10$ P $\backslash ).$ Let there be two firms producing sparkling water, each with a cost function of $\backslash ($ C$=2$ Q $\backslash )$ where the quantities produced by each firm are denoted as $\backslash ($ q$\_\{1\}\backslash )$ and $\backslash ($ q$\_\{2\}\backslash ).$

$1.$ What are the market equilibrium price and quantity when each firm behaves as a Cournot duopolist? What profit does each firm earn? $(5$ points$)$

$2.$ What are the market equilibrium price and quantity when each firm behaves as a Bertrand duopolist? What profit does each firm earn now? $(5$ points$)$

$3.$ Suppose that firm $1$ becomes the leader and chooses quantity first. Find the quantities of each firm, the market price, and the profits of each firm. What difference do you observe when the results are compared to those from part $(1)?(10$ points$)$