FLT near miss

Since Euler, we know that Diophantine equation $x^3+y^3=z^3$ has no solution.

However, Ramanujan found an infinite set of solutions for equation $x^3+y^3=z^3\pm 1$

Here is his stunning solution. Let :
$X_1=[9,10,12]$
$X_2=[791,812,1010]$
$X_3=[65601,67402,83802]$

And define $X_{n+3}=82\times X_{n+2}+82\times X_{n+1}-X_n$

Let $x_n=X_n[1],y_n=X_n[2],z_n=X_n[3]$

We have $x_n^3+y_n^3=z_n^3+(-1{)}^{n+1}$

For instance, $X_4=[5444135,5593538,6954572]\text{ and indeed }{5444135}^3+{5593538}^3={6954572}^3-1$

Find solution for index $123456789101112$

Format answer: $x,y,z\text{ modulo }{10}^{15}$

[My timing: instant]