$($Challenging$)$ A bank offers $2\mathrm{.}7\%$ p$.$a$.$ payable monthly on amounts less than $$30,000,4\mathrm{.}2\%$ p$.$a$.$ payable quarterly on amounts between $$30,000$ and $$50,000,$ and $6\mathrm{.}1\%$ p$.$a$.$ payable half$-$yearly on amounts of $$50,000$ or more. Interest is payable at the lowest monthly rate until the end of the month when the balance $($including accumulated interest$)$ first reaches $$30,000$ or more. Interest is payable at the lower quarterly rate until the end of the quarter when the balance $($including accumulated interest$)$ first reaches $$50,000$ or more. Thereafter the highest half$-$yearly rate applies. What is the amount required now so that

there is $$60,000$ in the account in twenty$-$two years$$ time? How many years does it take to obtain the highest half$-$yearly rate?