A differentiable function \(f\) is defined on the positive real numbers such that

\[\int_{1}^{xy} f(t) \, dt = y\int_{1}^{x} f(t) \, dt + x\int_{1}^{y} f(t) \, dt. \]

If \(f(1) = 3\), what is \(f(e)\)?

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