= Solution
Write the <Ehrenfeucht-Mostowski model> as the Skolem hull of its order-indiscernible skeleton $(a_i)_{i\in\eta}$. Every element is $t(a_{i_1},\ldots,a_{i_r})$ for a Skolem term $t$. Choose supports for the elements of $X$ and let $J\subseteq\eta$ be their union. Then
$$
|J|\leq |L|+|X|.
$$
When $\eta$ is well ordered, the type over $X$ of $t(a_{i_1},\ldots,a_{i_r})$ is determined by $t$ and the finite order pattern of the indices $i_k$ relative to $J$. There are at most $|L|$ terms and at most $|J|$ such finite patterns. Therefore the number of realized complete one-types is at most
$$
\boxed{|L|+|X|}.
$$
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