Let $k$ be a field. Consider two affine lines $$U_1 = \text{Spec}\ k[s] \ \ \text{and} \ \ U_2 = \text{Spec}\ k[t]$$ over $k$. Let $p_1$ and $p_2$ be the origins on $U_1$ and $U_2$ respectively. Construct a scheme $X$ by gluing $U_1$ and $U_2$ along $$V_1 = U_1 - \{p_1\} \ \ \text{and} \ \ V_2 = U_2 - \{p_2\}$$ via the isomorphism $V_1 \to V_2$ induced by the ring map $$k[t,t^{-1}] \to k[s,s^{-1}], \quad t \mapsto s.$$ Then $X$ is a non-separated non-affine smooth scheme over $k$, and is known as an affine line with a double origin.

What is the Chow ring $A(X)$ of $X$?

PS1: To be precise, the Chow ring $A(Y)$ for a smooth scheme $Y$ over $k$ is the one defined in Chapter 8 in Fulton's Intersection Theory.

PS2: The description of the question above is made more explicit thanks to a comment of Evgeny Shinder.

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