<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 For any positive integer $n$, let $a_n$ be the $y$-coordinate of the intersection point between the tangent line of the curve $y=x^n(1-x)$ at $x=2$ and the $y$-axis in the Cartesian coordinate system. Calculate the sum of the first 10 terms of the sequence $\{\log_2 \frac{a_n}{n+1}\}$. Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
55
For any positive integer $n$, let $a_n$ be the $y$-coordinate of the intersection point between the tangent line of the curve $y=x^n(1-x)$ at $x=2$ and the $y$-axis in the Cartesian coordinate system. Calculate the sum of the first 10 terms of the sequence $\{\log_2 \frac{a_n}{n+1}\}$.
test