Subharthi's minimum

Algebra Level 5

As \(x\) ranges over all real numbers, what is the minimum value of

\[ \left[ \sqrt{x^2 - 4x + 7 - 2\sqrt{2}} + \sqrt{x^2 - 8x + 27 - 6\sqrt{2}} \right]^4? \]

This problem is posed by Subharthi C.

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