Simplify $\frac{1+\sqrt{2}}{2+\sqrt{3}}$. Your solution can be converted to the form $A(1+\sqrt{B})-(\sqrt{C}+\sqrt{D})$, where A, B, C, and D are positive integers. What is $A+B+C+D$?
The number is 3(1 + sqrt(3)) - (sqrt(2) + sqrt(6)), so the answer is 3 + 3 + 2 + 6 = 14.
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