Consider a stock valued \(S_{0}=\$ 180\) at the beginning of the year. At the end of the year, its value \(S_{1}\) can be either \(\$ 152\) or \(\$ 203\), and the risk-free interest rate is \(r=3 \%\) per year. Given a put option with strike price \(K\) on this underlying asset, find the value of \(K\) for which the price of the option at the beginning of the year is equal to the intrinsic option payoff. This value of \(K\) is called the break-even strike price. In other words, the break-even price is the value of \(K\) for which immediate exercise of the option is equivalent to holding the option until maturity.

How would a decrease in the interest rate \(r\) affect this break-even strike price?