\[f(x) = \sqrt{x^{2} + 121} + \sqrt{x^{2} - 14x + 218}\]

Determine the value of \(x\) that minimizes the function above.

If \(x = \dfrac{a}{b},\) where \(a\) and \(b\) are positive coprime integers, then find \(a + b.\)

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