Are there any methods to "solve" large systems of boolean equations?

$$x_{i1}\vee x_{i2}\vee x_{i3} = b_i, \quad\text{for}\quad i=1,\dots,N,$$ where $x_i, b_i \in\{0, 1\}$

For example $$x_{1}\vee x_{2}\vee x_{5} = 0$$ $$x_{1}\vee x_{3}\vee x_{4} = 1$$ $$x_{2}\vee x_{3}\vee x_{5} = 1$$ $$x_{1}\vee x_{4}\vee x_{5} = 1$$ $$x_{1}\vee x_{2}\vee x_{3} = 0$$ $$...$$

The problem is to find $x$, that **satisfies the most of equations**. I couldn't employ any maxsat-like method, or integer linear programming to solve this problem.

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