Exercise $3-$ Collusion, punsishment and side payments

Consider a duopoly market with inverse demand given by

$\backslash [$

P$=180-3$ Q

$\backslash ]$

Firm $1$ has marginal cost of $12$ and firm $2$ has marginal cost of $36.$ There are no fixed costs. Both firms adopt a trigger strategy. The punishment price for firm $1$ is a price slightly below the marginal cost of firm $2.$ The punishment price of firm $2$ is its own marginal cost. It is assumed that firms produce an equal share of the output when the collusive price is being charged.

a What price would firm $1$ charge if it was the monopolist in this market? What price would firm $2$ charge if it was the monopolist in this market?

b Suppose the collusive price is equal to $\backslash ($ P$=108\backslash ).$ For which values of the discount factor $\backslash (\backslash$delta $\backslash )$ does the trigger strategy described above constitute a subgame$-$perfect Nash equilibrium?

c Suppose the collusive price is equal to $\backslash ($ P$=96\backslash ).$ For which values of the discount factor $\backslash (\backslash$delta $\backslash )$ does the trigger strategy described above constitute a subgame$-$perfect Nash equilibrium?

d Given your answers to $\backslash ($ b $\backslash ))$ and $\backslash ($ c $\backslash )),$ for which collusive price is the cartel more stable: $\backslash ($ P$=108\backslash )$ or $\backslash ($ P$=96\backslash )?$

e Now assume that, as in question $\backslash ($ b $\backslash )),$ the collusive price is set at $\backslash ($ P$=108\backslash ).$ Which minimal side payment $\backslash ($ A $\backslash )$ should firm $2$ make to firm $1$ in order for the cartel to be stable for all discount factors $\backslash (\backslash$delta $\backslash$geq $0\mathrm{.}625\backslash )?$