Three. (25 points) Given that $a, b, c$ are real numbers. The functions are $y_{1} = a x^{2} + b x + c, y_{2} = a x + b (a > 0)$. When $-1 \leqslant x \leqslant 1$, it is given that $-1 \leqslant y_{1} \leqslant 1$ and $y_{2}$ has a maximum value of 2. Try to find the area of the figure formed by connecting in sequence all the lattice points (points with integer coordinates) within and on the boundary of the closed region enclosed by the parabola $y_{1} = a x^{2} + b x + c$ and the line $y_{2} = a x + b$.