A Fibonacci Set of Sticks

Geometry Level 3

I was cleaning up my attic recently and found a set of at least 14 sticks which a curious Italian gentleman sold me some years ago. Trying hard to figure out why I bought it from him, I realized that the set has the incredible property that there are no \(3\) sticks that can form a triangle. If the set has two sticks of length \(1\), which are the smallest, what is the least possible length of the \({ 14 }^\text{th}\) stick?

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