Let $b=ra.$ Then your construction gives $$\mu(A_1\cap A_2\cap A_3)=0,\frac{(5r-2)^2}{8r}a \text{ or }(2r-1)a$$according as $r\le\frac25,\frac25\le r \le \frac23$ or $\frac23 \le r.$
Aaron Meyerowitz
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