Find the real number $x$ represented by continued fraction $[12;2,2,12,2,2,12,2,2,12\dots]$

Question

I need to fins the real vlaue of x for the continued fraction (Image attached)

I have tried partial coefficient method, but didn't get the exact answer.

I there any way where we can identify the value of x algebrically?

Answer 1

Hint:

Try to solve $$x=12+\dfrac{1}{2+ \dfrac{1}{2+ \dfrac{1}{x} } }$$