Let $a$ and $b$ be positive real numbers with $a \ge b$. Let $\rho$ be the maximum possible value of $\frac{a}{b}$ for which the system of equations  
$$a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2$$  
has a solution in $(x,y)$ satisfying $0 \le x < a$ and $0 \le y < b$. Then $\rho^2$ can be expressed as a fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let's think step by step and output the final answer within \boxed{}.
user