Bounding Correlation

I need to find min/max possible values of $$q$$ in the following situation.

"Consider random variables $$x, y$$, and $$z$$. Suppose it is known that $$\operatorname{Corr}[x, y]=\operatorname{Corr}[x, z]=$$ $$0.5$$. Denote $$q=\operatorname{Corr}[y, z]$$."

I can easily see that max is $$1$$ if Y = Z. However, can't find how to approach this problem to find minimum. Any hint?

Let $X,Y,Z$ be three random variables such that the correlation coefficients $\rho_{XY}=0.2, \rho_{YZ}=0.2$, what values can $\rho_{XZ}$ take?
You re right about the upper bound which is $$1$$. The constraint on lower bound is the covatiance matrix of the 3 variables shall be (semi) positive definite.
More specifically, normalizing variables won't change their correlation or positive definiteness of the covariance, so we consider the normalized variables $$x',y'z'$$. $$Cov([x',y',z'])= \begin{bmatrix} 1&0.5&0.5\\ 0.5&1&q\\ 0.5&q&1\\ \end{bmatrix}$$ By Generalized Sylvester Theorem we need to ensure all principal minor to be non-negative definite. $$1-q^2\geq 0 \\ \det Cov=1-q^2+\frac 12(q-1)\geq 0$$ You get $$-0.5\leq q\leq 1$$