Let the area enclosed by the lines $x+y=2, y=0, x=0$ and the curve $f(x)=\min \left\{x^2+\frac{3}{4}, 1+[x]\right\}$ where $[x]$ denotes the greatest integer $\leq x$, be $A$. Then the value of $12 A$ is $\_\_\_\_$ .
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