The effect of negative externalities on the optimal quantity of consumption

Consider the market for pharmaceuticals. Suppose that a pharmaceutical factory dumps toxic waste into a nearby river, creating a negative externality for those living downstream from the factory. Producing additional pharmaceuticals imposes a constant per$-$unit external cost of $$420.$ The following graph shows the demand $($private value$)$ curve and the supply $($private cost$)$ curve for pharmaceuticals.

Use the purple points $($diamond symbol$)$ to plot the social cost curve when the external cost is $$420$ per unit.

The market equilibrium quantity is $$ units of pharmaceuticals, but the socially optimal quantity of pharmaceuticals production is

$$ units.

To create an incentive for the firm to produce the socially optimal quantity of pharmaceuticals, the government could impose a $$ of

$$ per unit of pharmaceuticals. Please provide a graph and answers to the question