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Maximal cardinality of an independent subset $K^{\mathbb{N}}$, two distinct elements of which agree on at most finitely many entries

Let $K$ be a field, say infinite, and denote by $L$ the $K$-vector space $K^{\mathbb{N}}$. What is the maximal cardinality of a $K$-linearly independent subset $X$ of $L$ such that any two distinct elements of $X$ agree on at most finitely many elements of $\mathbb{N}$?