2011 HMMT November General Round Problem 10

Let r_1, r_2, \ldots, r_7 be the distinct complex roots of the polynomial P(x) = x^7 - 7. Let K = \prod_{1 \le i < j \le 7} (r_i + r_j), that is, the product of all numbers of the form r_i + r_j, where i and j are integers for which 1 \le i < j \le 7. Determine the value of K^2.

Answers are checked by value, so any equivalent form is accepted: 1/2, \frac{1}{2} and 0.5 all count as the same answer.

Full credit goes to HMMT for authoring these problems. This problem is from the November 2011 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.