The Stacks project

Exercise 111.54.4. Let $k$ be a field. Let $X$ be a “triangle” over $k$, i.e., you get $X$ by glueing three copies of $\mathbf{A}^1_ k$ to each other by identifying $0$ on the first copy to $1$ on the second copy, $0$ on the second copy to $1$ on the third copy, and $0$ on the third copy to $1$ on the first copy. It turns out that $X$ is isomorphic to $\mathop{\mathrm{Spec}}(k[x, y]/(xy(x + y + 1)))$; feel free to use this. Compute the Picard group of $X$.