The price of a European call that expires in six months and has a strike price of $$47$ is $$2\mathrm{.}17.$ The underlying stock price is $$46,$ and a dividend of $$1$ is expected in two months. The term structure is flat, with all risk$-$free interest rates being $5\%$ with continuous compounding.

a$.$ What is the price of a European put option that expires in six months and has a strike price of $$47?$

b$.$ Let us assume some mispricing now. Show in detail the arbitrage strategies and the arbitrage profit for the following two scenarios for the European put option price in tabular form:

$1)$ Scenario $1$: The European put price is $$4\mathrm{.}0.$

$2)$ Scenario $2$: The European put price is $$2\mathrm{.}5.$