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I want to write a sage math code to find the coefficient of $x^{15}d^7r^8$ in the expansion of $$\frac{1}{1-\bigg( \frac{xd+(xd)^2+(xd)^3}{1+(xd)+(xd)^2+(xd)^3} + \frac{xr+(xr)^2+(xr)^3}{1+xr+(xr)^2+(xr)^3} \bigg)}$$

I use free version of ChatGPT,but it writes always wrong code because it knows SAGE until 2021. Can you help me ?