A new surgical procedure reduces recovery time necessary after an operation. The new procedure costs $$25,000,$ and after undergoing the surgery there's a $50\%$ chance that a patient will live $1$ year and a $50\%$ chance they will live $9$ years. The QALY weight after receiving the surgery is $0\mathrm{.}6.$

The old procedure costs $$75,000$ and an average patient has a $50\%$ chance of surviving $2$ years and a $50\%$ chance of surviving $7$ years. The QALY weight after receiving the old surgery is $0\mathrm{.}6.$

Assume the value of one quality$-$adjusted life$-$year is $$100,000,$ and ignore discounting and consumption costs.

Based on a cost$-$benefit analysis, would you recommend that medical providers switch to the new technology? Please show all calculations.