The current spot price of a stock is $$34,$ the expected rate of return of the stock is $8\%,$ and the volatility of the stock is $20\%.$ The risk$-$free rate is $3\%.$ Compute the price of a derivative whose payoff in $6$ months is

$$8$ if the stock price in $6$ months, $S_{\frac{6}{12}},$ is below $$35,$

$$5$ if $35S_{\frac{6}{12}}55,$ and

nothing otherwise.