For an n-tuple of integers t = (a₁, ..., a_n), let (x₁, ..., x_n) be the solutions of the polynomial equation xⁿ + a₁x^n-1 + a₂x^n-2 + ... + a_n-1x + a_n = 0.

Consider the following two conditions:

• x₁, ..., x_n are all real.
• If x₁, ..., x_n are sorted, ⌊x_i⌋ = i for 1 ≤ i ≤ n. (⌊·⌋: 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 ∑S(t) = 2087 for all n-tuples t which satisfy both conditions.

Find ∑S(t) for n = 7.