Trisectors in a triangle

Geometry Level 3

Triangle \(ABC\) in the diagram above (not drawn to scale) has side lengths \(BC=a\) and \(CA=b\) with \(a<b.\) Line segments \(CD\) with length \(y\) and \(CE\) with length \(x\) trisect \(\angle BCA.\)

Given that \(y=\frac{2}{3}x,\) find the smallest value of \(k\) such that \(\frac{a}{b}<k,\) to two decimal places.

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