= Solution
For the classical XY ferromagnet choose every <spin> parallel within the $xy$ plane. Each nearest-neighbor and next-nearest-neighbor in-plane dot product is $S^2$. The printed sum counts one bond of each range per <spin>, so
$$
\boxed{E_{\rm F}=-S^2(J_1+J_2).}
$$
There is no additional factor of two from counting both directions: the <Hamiltonian operator> already lists only $n\to n+1$ and $n\to n+2$. Alignment along $z$ would give zero XY exchange energy and is not the in-plane ordered state used for these expressions.
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