Given that the real number ABC is not equal, a + 1 / b = B + 1 / C = C + 1 / D = D + 1 / a = x, find the value of X

Given that the real number ABC is not equal, a + 1 / b = B + 1 / C = C + 1 / D = D + 1 / a = x, find the value of X

B = 1 / (x-a), substituting B + 1 / C = x, we get the following formula:
c=(x-a)/(x^2-ax-1)
Substitute C + 1 / D = x to get (x-a) / (x ^ 2-ax-1) + 1 / D = X
DX ^ 3 - (AD + 1) x ^ 2 + (a-2d) x + (AD + 1) = 0
From D + 1 / a = x, so AD + 1 = ax, so (D-A) x ^ 3 + (a-2d) x + AX = 0
So (D-A) x ^ 3 + 2 (A-D) x = 0
Because a ≠ D, x ^ 3-2x = 0
So x = 0 or √ 2 or - √ 2
When x = 0, ab = BC = CD = ad = - 1, then a = C, B = D
To sum up, x = ± √ 2