Fibonacci Divisibility Graph

Let \(\{F_i\}_{i=1}^\infty\) be the sequence of Fibonacci numbers; that is, \(F_1 = F_2 = 1\) and \(F_{n+2} = F_{n+1} + F_n\) for all \(n \ge 1\).

Consider a graph with vertices labeled \(F_3, F_4, F_5, \ldots, F_N\), and two (different) vertices \(x,y\) are connected by an edge if and only if \(x|y\) or \(y|x\). The above picture is the graph for \(N = 12\).

Find the largest \(N\) such that this graph is planar. If there is no largest \(N\), enter \(0\).

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