On October $22$nd

$\text{'},$ a January put option on a stock with a strike price of $$55$

costs $$2.$ The current stock price is $$56.$ A trader buys the put option and holds it until January $($the maturity day$).$ Suppose the trader only holds the option.

$1)$ Under what circumstances will the option be exercised at the maturity of the option?

$2)$ Under what circumstances will the holder of the option make a gain at the maturity of the option?