Find $\displaystyle{ \frac{2}{1 + 2\sqrt{3}} + \frac{3}{2 - \sqrt{3}}}$, and write your answer in the form $\displaystyle \frac{A + B\sqrt{3}}{C}$, with the fraction in lowest terms and $A > 0$. What is $A+B+C$?
<|im_start|>system
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<|im_start|>user
Find $\displaystyle{ \frac{2}{1 + 2\sqrt{3}} + \frac{3}{2 - \sqrt{3}}}$, and write your answer in the form $\displaystyle \frac{A + B\sqrt{3}}{C}$, with the fraction in lowest terms and $A > 0$. What is $A+B+C$?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
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Find $\displaystyle{ \frac{2}{1 + 2\sqrt{3}} + \frac{3}{2 - \sqrt{3}}}$, and write your answer in the form $\displaystyle \frac{A + B\sqrt{3}}{C}$, with the fraction in lowest terms and $A > 0$. What is $A+B+C$?
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