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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