Mathematica Mathematica: solving system of nonlinear equations

[gothmogsbane]

I am trying to solve the following system of 2n variables:

w₁ + w₂ + ... + w_n = b₀
w₁x₁ + w₂x₂ + ... + w_nx_n = b₁
w₁x₁² + w₂x₂² + ... + w_nx_n² = b₂
...
w₁x₁^2n-1 + w₂x₂^2n-1 + ... + w_nx_n^2n-1 = b_2n-1

for w₁, w₂ ... w_n and x₁, x₂ ... x_n.

The problem is the using the Solve command returns n! solutions, because the x_i are free to switch positions. For example, using n=2 and b_i=1/(i+1):

{{w1 -> 1/2, w2 -> 1/2, x2 -> 1/6*(3 - Sqrt[3]), x1 -> 1/6*(3 + Sqrt[3])},
{w1 -> 1/2, w2 -> 1/2, x2 -> 1/6*(3 + Sqrt[3]), x1 -> 1/6*(3 - Sqrt[3])}}

Is there any way to force Mathematica to make the assumption that x₁ <= x₂ <= ... <= x_n in order to return only one solution?

 

[FunkyDwarf]

I think you need to use Reduce and Assumptions from memory