Let \(f(x, y) = \frac{\sum_{a_0 = 0}^y \sum_{a_1 = 0}^y \ldots \sum_{a_x = 0}^y \prod_{i = 0} ^x a_i}{\Big(\frac{x(x+1)}{2}\Big)^{y+1}}\)

Find \(\text{min}\Big(\frac{f(4, y)}{f(19, y)}\Big)\).

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