= Solution
The <Einstein–Podolsky–Rosen criterion of reality> says that if a physical quantity can be predicted with certainty without disturbing a system, then an element of physical reality corresponds to that quantity.
Let $X_i,Y_i$ be Pauli measurements on particle $i$. The <GHZ state> in the question obeys
$$
X_1Y_2Y_3=+1,\qquad
Y_1X_2Y_3=+1,\qquad
Y_1Y_2X_3=+1
$$
with certainty. Each local outcome can therefore be predicted by spacelike-separated measurements on the other two particles. The EPR criterion assigns predetermined values $x_i,y_i\in\{-1,1\}$ satisfying those three equations. Multiplying them and using $y_i^2=1$ gives
$$
x_1x_2x_3=+1.
$$
Quantum theory instead predicts
$$
X_1X_2X_3|\psi_{\rm GHZ}\rangle=-|\psi_{\rm GHZ}\rangle,
$$
so a joint $X$ measurement has product $-1$ with certainty. The EPR elements of reality therefore contradict the quantum prediction, which is the <GHZ theorem> form of <Bell theorem>.
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