The function $f : \mathbb{R} \to \mathbb{R}$ satisfies the equation \( f(x^2) f''(x) = f'(x) f'(x^2) \) for all real numbers \( x \). Given that \( f(1) = 1 \) and \( f'''(1) = 8 \), determine the value of \( f'(1) + f''(1) \).

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