Continuable fractions

Number Theory Level 1

Let XX and YY be positive integers such that the first few digits of the fractions 2970, 70169, 169408, 408X, XY \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 X+Y?{{\color{#624F41}{X}}}+{{\color{grey}{Y}}}?

Note: The fractions are not equal.

Pi Han Goh by Pi Han Goh
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