For My Friend Nihar Mahajan (Happy Birthday Man)

Algebra Level 5

$\large{{ x }_{ n+1 }=\left\lfloor \sqrt [ 3 ]{ { x }_{ n }^{ 3 }-\left\lfloor \sqrt [ 3 ]{ { x }_{ n } } \right\rfloor } \right\rfloor }$

Let there be a sequence defined by above rule for $$n \ge 1$$. For $$n \ge k$$ the value of $$a_{n}$$ will be $$0$$, for some positive integer $$k$$. Find the value of $$k$$, if $$a_{1}=839475687$$.

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