Correlation Range with Four Variables

Linear Algebra · Hard · Free problem
You have four random variables $a$, $b$, $c$, and $d$. Every pairwise correlation equals $\frac{1}{2}$, except for $\text{Corr}(a, d)$, which is unknown. Call it $\rho$. What is the range of valid values for $\rho$? In other words, for which values of $\rho$ does a valid joint distribution exist?

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