Raven Construction Company began a long$-$term contract in $2022.$ The contract price was $$600,000.$ The estimated cost of the contract at the time it was begun was $$450,000.$ The actual cost incurred in $2022$ was $$300,000.$ The contract was completed in $2023,$ and the cost incurred that year was $$180,000.$ Under the percentage of completion method:

a$.$ Raven should report $$300,000$ of income in $2022.$

b$.$ Raven should report a $$30,000$ loss in $2023.$

c$.$ Raven should report $$60,000$ profit on the contract in $2023.$

d$.$ Raven will receive interest $($under the lookback method$)$ on the overpayment of taxes in $2022.$

e$.$ Raven must pay interest $($under the lookback method$)$ on the overpayment of taxes in $2022.$