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Calculus Level 3

If \(f(x) = x^2 + x g'(1) + g''(2)\) and \(g(x) = f(1)x^2 + xf'(x) + f''(x) ~\forall ~x\in \mathbb{R}\), and if \(f(x)\) is in the form of \(ax^2 + bx\) and \(g'(x) = -c\), find the value of \(a + 2b + 3c\).

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