Given that \(x, y,\) and \(z\) are real numbers satisfying \(x + y + z = 10\) and \(x^2 + y^2 + z^2 = 50\), find the maximum possible value of \((x + 2y + 3z)^2 + (y + 2z + 3x)^2 + (z + 2x + 3y)^2\).

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