Let $\min (x, y)$ denote the smaller of the numbers $x$ and $y$. For what $\lambda$'s does the equation  
$$\int_{0}^{1} \min (x, y) f(y) d y=\lambda f(x)$$  
have continuous solutions which do not vanish identically in ( 0,1$)$? The largest $\lambda$ can be expressed in the form of $\frac{m}{n\pi^{2}}$. Find the value of $m + n$.

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