On Jan. $1,2022.$ You entered a one$-$year long forward contract on a stock paying $$0\mathrm{.}5$ at the end of every quarter. The stock price was at $$50$ when you initiated the forward contract, and the risk$-$free rate is $10\%$ per annum with continuous compounding

$($a$)$ What is the fair forward price and the initial value of the forward contract?

$($b$)$ Ten months later, the price of the stock decreases to $$45$ and the risk$-$free rate is still $10\%.$ What is the forward price and the value of the forward contract you longed?

$($c$)$ Suppose on Jan. $1.2022,$ the $1-$year forward's market price is $$50,$ is there any arbitrage opportunity? Be specific about your trading strategy in terms of the action you need to take and the arbitrage profit.