We define an $S$-number to be a natural number, $n$, that is a perfect square and its square root can be obtained by splitting the decimal representation of $n$ into 2 or more numbers then adding the numbers.

For example, 81 is an $S$-number because $\sqrt{81} = 8+1$.

6724 is an $S$-number: $\sqrt{6724} = 6+72+4$.

8281 is an $S$-number: $\sqrt{8281} = 8+2+81 = 82+8+1$.

9801 is an $S$-number: $\sqrt{9801}=98+0+1$.

Further we define $T(N)$ to be the sum of all $S$ numbers $n\le N$. You are given $T(10^4) = 41333$.

Find $T(10^{12})$