How to represent $\limsup \cdot \liminf$ of Booleans

Let $X$ be some set, and let $A,B\subset X$. By $1_A(x)$ let us mean the indicator/characteristic function. Let $(x_n)_{n\in \Bbb N}$ be some sequence in $X$. I have an expression of the form $$\left(\limsup_n 1_A(x_n)\right)\cdot\left(\liminf_n 1_B(x_n)\right) \tag{1}$$ which equals $1$ exactly when $(x_n)_n$ visits $A$ infinitely often, and $B^c$ only finitely often. It is possible to rewrite $(1)$ in a simpler shape, e.g. something like $\limsup f(x_n)$ where $f$ is not necessary an indicator?

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