$|$ In a Lean Manufacturing environment, a production line processes $2,000$ items daily. If the defect rate is $5\%,$ how many defects should be expected? Additionally, if the cost to rectify each defect is $$10$ and the process improvement initiative aims to reduce the defect rate to $3\%,$ calculate the expected daily cost savings from this improvement. Also, if the process improvement initiative costs $$2,000$ to implement, how many days will it take for the savings to offset the implementation cost? $($Show Work$)$