What is the minimum value of \( \sqrt{x^2 - 4x + 7 - 2\sqrt{2}} + \sqrt{x^2 - 8x + 27 - 6\sqrt{2}} \) where \( x \) is a real number? The original answer is in the form \( k\sqrt{2} \), please find the value of \( k \).

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