I'm trying to know if there is an efficient way to find the smallest (i.e lexicographically) triplet $(a,b,c)$ of integers verifying $$a^2+b^2+c^2 = x$$ $$a^3+b^3+c^3=y$$ $$a^4+b^4 + c^4 = z$$ if $(x,y,z)$ is known.
We assume that a solution exists for that triplet $(x,y,z)$.
Originally, this question is part of an algorithmic problem. So what I want is a fast way to find the triplet without having to brute-force.
What I've tried (the brute-force approach): going through all the possible values of $a$ since we can deduce an upper bound for its value, we remain with three equations where it's easy to find $b$ and $c$ and see if they are integers.
EDIT: We may assume as well that $a<b<c$.