Tribonacci Treat

\[ \begin{eqnarray} \displaystyle \lim_{n \to \infty} \frac {F_{n+1}}{F_n} & = & \frac {A + \sqrt B }{C} \\ \displaystyle \lim_{n \to \infty} \frac {T_{n+1}}{T_n} & = & \frac {1}{D} \left ( E + \sqrt[3]{F- G \sqrt H} + \sqrt[3]{I + J \sqrt K } \right) \\ \end{eqnarray} \]

Let the \(n^\text{th} \) term of a Fibonacci sequence and Tribonacci sequence be denoted as \(F_n\) and \(T_n\) respectively.

In its simplest form, we have positive integers \(A,B,C,D,E,F,G,H,I,J,K\) such that the limits above are satisfied.

Evaluate \(A+B+C+D+E+F+G+H+I+J+K\).

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