A function \(f\) is such that:

\[\large \begin{cases} yf(x) + x^2f(y) = f(xy) \\ \displaystyle \lim_{x \to 0} \frac{f(x)}{x} = 1 \end{cases} \]

Prove that \(f\) is differentiable infinitely many times.

Submit your answer as the value of \(f'(10) - 10 f''(20)\).

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