= Solution
The curve has <good reduction of an elliptic curve>[good reduction] when it admits a <Minimal Weierstrass equation> over $\mathcal O_K$ whose discriminant is a unit, so reduction modulo $\pi$ is a nonsingular elliptic curve $\widetilde E/k$. Write a point of $E(K)$ in primitive projective coordinates $(X:Y:Z)$ over $\mathcal O_K$. Reducing all three coordinates defines
$$
\operatorname{red}:E(K)\longrightarrow\widetilde E(k),
\qquad (X:Y:Z)\longmapsto(\widetilde X:\widetilde Y:\widetilde Z).
$$
Primitivity ensures that the reduced triple is not zero. The addition morphism on the smooth Weierstrass model reduces to the addition morphism on $\widetilde E$; equivalently, this follows from the <valuative criterion for properness> for the smooth proper group scheme. Therefore $\operatorname{red}(P+Q)=\operatorname{red}(P)+\operatorname{red}(Q)$.
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