For an $n$-tuple of integers $t=(a_1,\dots ,a_n)$, let $(x_1,\dots ,x_n)$ be the solutions of the polynomial equation $x^n+a_1{x}^{n-1}+a_2{x}^{n-2}+\cdots +a_{n-1}x+a_n=0$.

Consider the following two conditions:

• $x_1,\dots ,x_n$ are all real.
• If $x_1,\dots ,x_n$ are sorted, $\lfloor x_i\rfloor =i$ for $1\le i\le n$. ($\lfloor \cdot \rfloor$: floor function.)

In the case of $n=4$, there are $12$ $n$-tuples of integers which satisfy both conditions.
We define $S(t)$ as the sum of the absolute values of the integers in $t$.
For $n=4$ we can verify that $\sum S(t)=2087$ for all $n$-tuples $t$ which satisfy both conditions.

Find $\sum S(t)$ for $n=7$.