Let $k$ be a field with $\operatorname{char}(k) \neq 2,3$. Let $a,b \in k$ satisfy $4a^3 + 27b^2 \neq 0$, and let $E/k$ be the elliptic curve in short Weierstrass form
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\begin{align*}
E: y^2 = x^3 + ax + b
\end{align*}
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with identity element $O = [0:1:0]$. If $x_1 \in k$ and $P = (x_1,0) \in E(k)$ is an affine $k$-point, then