Alejandra Garcia is purchasing a new $2024$ Porsche $911$ at a cost of $$223,000.$ The dealership will finance the purchase of the car at $2\mathrm{.}9\%$ over $84$ months, or they will provide a $$10,000$ rebate if she pays cash.

a$.$ What is her payment if she takes the dealer's $2\mathrm{.}9\%$ financing?

b$.$ What is the true cost of the dealer's $2\mathrm{.}9\%$ financing?