TxO Math Bowl 2024 - Guts Contest - Set 7 Problem 2

Given

$x=3+\dfrac{2x^2-1}{3+\dfrac{2x^3-x}{3x+2x^2-1}}$

the largest real solution can be written as $\frac{\sqrt a - b}{c}$, where $a,b,$ and $c$ are positive integers with $a$ not divisible by the square of any prime. Compute $a+b+c$.