Continuable fractions

Number Theory Level 1

Let $X$ and $Y$ be positive integers such that the first few digits of the fractions $\dfrac{29}{{\color{#D61F06}{70}}},\ \dfrac{{\color{#D61F06}{70}}}{{\color{#3D99F6}{169}}},\ \dfrac{{\color{#3D99F6}{169}}}{{\color{#20A900}{408}}},\ \dfrac{{\color{#20A900}{408}}}{{\color{#624F41}{X}}},\ \dfrac {{\color{#624F41}{X}}}{{\color{grey}{Y}}}$ are all equal to 0.4142. What is ${{\color{#624F41}{X}}}+{{\color{grey}{Y}}}?$

Note: The fractions are not equal.