There are two companies in the market. Firm A has the advantage of being able to select profit$-$maximizing output before firm B$.$ The inverse demand function of the market is P $=500-2$Q$.$ And firm A$\text{'}$s cost function is C$($a$)=2$Qa$,$ and firm B$\text{'}$s cost function is C$($b$)=4$Qb$.$

$1.$ What are the profit$-$maximizing outputs of the two companies in the market?

$2.$ What is the equilibrium price of the market?

$3.$ How much profit can each of the two companies generate?

$4.$ Instead of considering antitrust matters, explain whether mineral water would be beneficial if it merged with the corn industry. If it is not beneficial, explain why, and specifically present how much the profit would be$,$ if so