The price of an American call on a non-dividend-paying stock is $$\$ 4$$. The stock price is $$\$ 31$$, the strike price is $$\$ 30$$, and the expiration date is in 3 months. The risk-free interest rate is $8 \%$. Derive upper and lower bounds for the price of an American put on the same stock with the same strike price and expiration date.