Continuable fractions

Let \(X\) and \(Y\) be positive integers such that the first few digits of the fractions \[ \dfrac{29}{{\color{red}{70}}},\ \dfrac{{\color{red}{70}}}{{\color{blue}{169}}},\ \dfrac{{\color{blue}{169}}}{{\color{green}{408}}},\ \dfrac{{\color{green}{408}}}{{\color{brown}{X}}},\ \dfrac {{\color{brown}{X}}}{{\color{grey}{Y}}} \] are all equal to 0.4142. What is \({{\color{brown}{X}}}+{{\color{grey}{Y}}}?\)


Note: The fractions are not equal.

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