Acute triangle $ABC$ satisfies $AB=2AC$ and $AB^4+BC^4+CA^4 = 2006 \cdot 10^{10}$. A tetrahedron $DEFP$ is formed by choosing points $D$, $E$, and $F$ on the segments $BC$, $CA$, and $AB$ (respectively) and folding $A$, $B$, $C$ over $EF$, $FD$, and $DE$ (respectively) to the common point $P$. Let $R$ denote the circumradius of $DEFP$. Compute the smallest positive integer $N$ for which we can be certain that $N \geq R$. It may be helpful to use $\sqrt[4]{1239} = 5.9329109\ldots$.

Let's think step by step and output the final answer within \boxed{}.
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