Feasible correlation range for a third pair

For three random variables $X$, $Y$, $Z$ we know $\rho_{YZ}=\tfrac{12}{13}$ and $\rho_{XY}=\tfrac{5}{13}$, where $\rho$ denotes correlation. The set of values $\rho_{XZ}$ can take turns out to be an interval $[0,b]$. Find $b$ as a reduced fraction.

$b=120/169$. Any correlation matrix must be positive semi-definite. With $\rho=\operatorname{Corr}(X,Z)$ the $3\times3$ determinant equals $\tfrac{120}{169}\rho-\rho^2$, which is non-negative exactly for $0\le\rho\le\tfrac{120}{169}$ (the leading $2\times2$ minor is already positive).