A stock price is currently $$60.$ Over each of the next two three$-$month periods it is expected to go up by $6\%$ or down by $5\%$ and in the next leap it will go up by $8\%$ and down by $6\%.$ The risk$-$free interest rate is $5\%$ per annum with continuous compounding. What is the value of a six$-$month European and American put option with a strike price of $$62?$