<|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
 Given $N=2^n$ (where $n\in\mathbb{N}^*, n\geq 2$), place $N$ numbers $x_1, x_2, \ldots, x_N$ sequentially into $N$ positions labeled from 1 to $N$, resulting in the permutation $P_0=x_1x_2\ldots x_N$. Extract the numbers located in the odd and even positions of this permutation and place them in order into the corresponding first $\frac{N}{2}$ and last $\frac{N}{2}$ positions, forming permutation $P_1=x_1x_3\ldots x_{N-1}x_2x_4\ldots x_N$. Let's call this operation a C transformation. Divide $P_1$ into two segments, each containing $\frac{N}{2}$ numbers, and perform a C transformation on each segment to obtain $P_2$. For $2\leq i\leq n-2$, divide $P_i$ into $2^i$ segments, each with $\frac{N}{2^i}$ numbers, and apply C transformations to each segment to get $P_{i+1}$. For example, when $N=8$, $P_2=x_1x_5x_3x_7x_2x_6x_4x_8$, and $x_7$ is in the fourth position in $P_2$. When $N=32$, find the position of $x_{21}$ in $P_3$. 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
7
Given $N=2^n$ (where $n\in\mathbb{N}^*, n\geq 2$), place $N$ numbers $x_1, x_2, \ldots, x_N$ sequentially into $N$ positions labeled from 1 to $N$, resulting in the permutation $P_0=x_1x_2\ldots x_N$. Extract the numbers located in the odd and even positions of this permutation and place them in order into the corresponding first $\frac{N}{2}$ and last $\frac{N}{2}$ positions, forming permutation $P_1=x_1x_3\ldots x_{N-1}x_2x_4\ldots x_N$. Let's call this operation a C transformation. Divide $P_1$ into two segments, each containing $\frac{N}{2}$ numbers, and perform a C transformation on each segment to obtain $P_2$. For $2\leq i\leq n-2$, divide $P_i$ into $2^i$ segments, each with $\frac{N}{2^i}$ numbers, and apply C transformations to each segment to get $P_{i+1}$. For example, when $N=8$, $P_2=x_1x_5x_3x_7x_2x_6x_4x_8$, and $x_7$ is in the fourth position in $P_2$. When $N=32$, find the position of $x_{21}$ in $P_3$.
test