\[ \sqrt[4] {x ^ 5 + 48 \sqrt[3] {3} - 96 \sqrt[3] {9}} = \sqrt[5] {x ^ 4 - 48 \sqrt[3] {3} + 96 \sqrt[3] {9}} \]

If real \(x\) satisfies the equation above, then find the value of the infinite root:

\[ \sqrt[3] {x ^ 3 + \sqrt[3] {x ^ 3 + \sqrt[3] {x ^ 3 + \cdots } } } \]

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