AIME 2025 I / A piecewise linear periodic function is defined by …

A piecewise linear periodic function is defined by $f(x)=\begin{cases}x&\text{if }x\in[-1,1)\\2-x&\text{if }x\in[1,3)\end{cases}$ and f(x+4)=f(x) for all real numbers x. The graph of f(x) has the sawtooth pattern. The parabola x=34y² intersects the graph of f(x) at finitely many points. The sum of the y-coordinates of these intersection points can be expressed in the form (a+b√(c))/(d), where a,b,c, and d are positive integers, a,b, and d have greatest common divisor equal to 1, and c is not divisible by the square of any prime. Find a+b+c+d.

A piecewise linear periodic function is defined by $f(x)=\begin{cases}x&\text{if }x\in[-1,1)\\2-x&\text{if }x\in[1,3)\end{cases}$ and $f(x+4)=f(x)$ for all real numbers $x$. The graph of $f(x)$ has the sawtooth pattern. The parabola $x=34y^2$ intersects the graph of $f(x)$ at finitely many points. The sum of the $y$-coordinates of these intersection points can be expressed in the form $\frac{a+b\sqrt{c}}{d}$, where $a,b,c,$ and $d$ are positive integers, $a,b,$ and $d$ have greatest common divisor equal to 1, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.