Costing and pricing methodologies

A new mineral, unobtanium, is discovered in only two locations on Earth, each owned by a different firm. It is a vital part of producing infant pacifiers, and has the interesting trait of often disappearing when needed. Unobtanium is the exact same regardless of where it is produced. Both firms face the market demand curve
P=1000-.5Q
And both firms have a constant marginal cost of 25. Assume the firms compete in a Cournot fashion.
Derive firm 1’s optimal response as a function of whatever quantity firm 2 will choose. Find Firm 2’s optimal response curve using symmetry

Solve for both firm’s individual quantities, as well as the market quantity and price

Examining your best response functions, explain why the firms would struggle to collude.

Derive the perfect competition price and quantity. Is demand more or less elastic at the perfect competition price or the Cournot duopoly price?

Assume that firm 1 is now able to move first.
Is there a first mover advantage? Explain how you know this referring to the first mover’s optimization problem.

What would happen if the two firms instead competed by choosing price?