$1.$ Jack is considering selling his elliptical machine, which he never really used, is high quality, and has been taking up space in his spare bedroom. He lists it on Facebook Marketplace for $"$$$1,200$ or best offer" but in reality won't sell it for less than $$1,000.$ Lina is looking to purchase a used elliptical machine. For a high$-$quality elliptical, she is willing to pay up to $$1,300,$ and for a low$-$quality elliptical, she is willing to pay $$600.$ Lina is risk neutral but she cannot tell if the elliptical machine is high quality or low quality. If she believes that $60\%$ of used elliptical machines are high quality and $40\%$ are low quality, what is the maximum price that Lina would be willing to pay? Would Jack agree to this price?