Fibonacci-like sequences

Conside The Fibonacci sequence is defined by \[f_1 = 1, f_2 = 1, f_n = f_{n-1} + f_{n-2} \text{ for } n \geq 3.\] We have that \(f_{13} = 233.\) Consider a sequence such that

\[g_1 = 1, g_2 = x, g_n = g_{n-1} + g_{n-2} \text{ for } n \geq 3.\]

Determine the sum of all positive integer values of \(x \geq 2\) for which \(233\) is a term of the sequence \(g\).

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