You're tasked with developing a robust and efficient pricing model for a long-term American put option, assuming the underlying asset follows a geometric Brownian motion and pays continuous dividends. Using the Finite Difference Method (FDM) to solve the relevant Partial Differential Equation (PDE), walk me through how you would construct and implement such a model. This should include your choice of discretization scheme (e.g., explicit, implicit, Crank-Nicolson), how you would handle boundary conditions, and most critically, the early exercise condition. What are the unique challenges and best practices for numerically incorporating early exercise?

你被要求为一种长期美式看跌期权（假设标的资产服从几何布朗运动，且支付连续股息）开发一个鲁棒且高效的定价模型。你需要使用有限差分法 (FDM) 来解决相关的偏微分方程 (PDE)。请你详细阐述如何构建和实现这样一个 FDM 模型，包括选择哪种差分方案、如何处理边界条件以及最关键的早期行权条件。特别地，你认为在数值实现中，处理早期行权有哪些独特挑战和最佳实践？