Solving the equations arising from a Lagrange multipliers problem

Find the maximum value of 2

x+3y+7z, subject to the constraint x^4+y^4+z^4= 1.

First I formed my Lagrangian for this problem:

L(x,y,z) = 2x + 3y + 7z - λ(x^4 + y^4 + z^4 -1)

I then took my partial derivatives:

dL/dx = 2 - 4λx^3

dL/dy = 3 - 4λy^3

dL/dz = 7 - 4λz^3

For each of these I made x,y and z the subject respectively, giving:

x^3 = 1/2λ ; y^3 = 3/4λ ; z^3 = 7/4λ

I now subbed these into my constraint. This gave a scary equation with the above λ expressions raised to the power of 4/3 being equal to 1. This is where I'm stuck, can anyone help me please?

I found a solution, but not in surd form, 2.38534589 satisfies the λ equation being equal to 1. But my answer must be in the form of a surd