algebra precalculus - Non-linear system of equations over the positive integers — more unknowns than equations

This exercise appeared on a german online tutoring board and caught my attention but stumbled me for hours. The task is to find 6 distinct positive three digit integers satisfying:

$x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}=4.198$
$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}=3.215.224$
$x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}+x_{5}^{3}+x_{6}^{3}=2.600.350.972$

According to the power mean inequality or Cauchy-Schwarz the numbers must lie relatively closely together. However a brief search lead nowhere.
For simplicity I set $4.198=A$, $3.215.224=B$ and $2.600.350.972=C$ and then my approach was to manipulate the three equations and perhaps use that no square is negative. For example
$6B-A^{2}=\sum_{iIf now $6B-A^{2}$ would result in something like
$5*1^{2}+4*2^{2}+3*3^{2}+2*4^{2}+1*5^{2}=105$

we could tell exactly what our $x$ were. Unfortunately it gives $1.668.140$ and we cannot conclude much.
Similar reasoning with factoring to $\sum_{iIf there exists such a factorization, my intuition says it would only make sense if the $x$ form an arithmetic sequence, otherwise we would get different factors on the right side that can't appear on the left side. (does this sound reasonable?) But substituting gives no solution. Also, I don't know what other, more sophisticated factorization would lead me to the solution.
I'm running out of ideas, how can this problem be solved?

Links to the original problem:
https://www.geocaching.com/geocache/GC69JE0_lotto-mal-anders
https://www.gutefrage.net/frage/matherechenart-gesucht-mehrere-variablen-mit-festen-ergebnis?foundIn=unknown_listing