Express $355$ as a sum of three squares. PROBLEM

Find all the ways to express $355$ as a sum of three squares.

MY ATTEMPT
By Legendre's three-square theorem, since $355$ is not of the form $n = {4^a}(8b+7)$ (for $a, b \in \mathbb{Z}$), then $355$ can be expressed as a sum of three squares.
WLOG, we can restrict to positive integral solutions in
$$x^2 + y^2 + z^2 = 355,$$
where $x \leq y \leq z$.
WolframAlpha gives
$$x = 7, y = 9, z = 15,$$
and
$$x = 3, y = 11, z = 15.$$
QUESTION

Are these all of the solutions?


I do not consider any order between $x,y,z$ here.
$x^2+y^2+z^2\equiv 1\pmod 3$ and since $(0,1,2)\xrightarrow{x^2}(0,1,1)$ then we can only be in the $0+0+1$ configuration, so two numbers are divisible by $3$.
$x=3a,\ y=3b$ gives $9(a^2+b^2)+z^2=355\iff z^2\equiv 4\pmod 9\iff z\equiv 2,7\pmod 9$
Since $19^2=361>355$ then only $z=2,7,11,16$ are possible.
For $a^2,b^2$ we are limited to values $0,1,4,9,16,25,36$ in the table below and it is easy to find them manually
$\begin{array}{|c|c|c|c|}\hline
z & \frac{355-z^2}9 & a^2+b^2 & x,y\\\hline
2 & 39 & \varnothing \\
7 & 34 & 9+25 & 9,15\\
11 & 26 & 1+25 & 3,15\\
16 & 11 & \varnothing \\\hline
\end{array}$
