On January $1,2023,$ Holiday Corporation completed the following transactions $($you can assume a $6\%$ annual interest rate$)$:

a$.$ Bought equipment and agreed to pay $$182,000$ at the end of three years.

b$.$ Established an investment account by depositing a single amount that will increase to $$95,000$ at the end of six years.

c$.$ Rented a robotic packaging machine and was given the option of paying $$35,000$ at the end of each of the next three years or paying $$90,000$ immediately.

Required:

a$.$ In $($a$),$ what is the cost of the equipment that should be recorded at the time of the purchase?

b$.$ In $($b$),$ what single amount needs to be deposited in this account on January $1,2023?$

c$.$ In $($c$),$ which option for the robotic packaging machine would the company select?