<s>[INST] A five-digit palindrome number has the form $abcde$.

\begin{align*}
S&= \sum_{\substack{a\neq 0,\\ a,b,c,d,e\text{ distinct}}} a^2+b^2+c^2+c^2+b^2+a^2 \\
&=(a^2+b^2+c^2)\cdot 4 + 2c^2 \\
&= 4\sum_{a\neq 0} a^2 + 4\cdot9\cdot10\cdot2+2\sum_{c=1}^9c^2\cdot132 \\
&= 4\cdot \frac{9\cdot10\cdot19}{6} \cdot132+4\cdot 9\cdot 2\cdot 10+2\cdot 330\cdot 4\cdot 9\\
&=4\cdot330\cdot19+720+4\cdot 5940\\
&=4(6270+5940) +720\\
&= \boxed{48480}.
\end{align*}How many five-digit palindromes exist?[/INST]