Given that $\alpha$ and $\beta$ are the roots of $x^2 - 2x - 1 = 0,$ find $5 \alpha^4 + 12 \beta^3.$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Given that $\alpha$ and $\beta$ are the roots of $x^2 - 2x - 1 = 0,$ find $5 \alpha^4 + 12 \beta^3.$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
169
Given that $\alpha$ and $\beta$ are the roots of $x^2 - 2x - 1 = 0,$ find $5 \alpha^4 + 12 \beta^3.$
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