
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let $N=2^{n}(n\in\mathbb{N}^{*},n\geqslant 2)$, and place $N$ numbers $x_{1}, x_{2}, \ldots, x_{N}$ sequentially into $N$ positions numbered $1, 2, \ldots, N$, to get the permutation $P_{0}=x_{1}x_{2}\ldots x_{N}$. Take out the numbers located at odd and even positions from this permutation, and place them in order into the corresponding first $\dfrac{N}{2}$ and the latter $\dfrac{N}{2}$ positions, to get the permutation $P_{1}=x_{1}x_{3}\ldots x_{N-1}x_{2}x_{4}\ldots x_{N}$. This operation is called the $C$ transformation. Divide $P_{1}$ into two parts, each with $\dfrac{N}{2}$ numbers, and perform the $C$ transformation on each part to get $P_{2}$. When $2\leqslant i\leqslant n-2$, divide $P_{i}$ into $2^{i}$ parts, each with $\dfrac{N}{2^{i}}$ numbers, and perform the $C$ transformation on each part to get $P_{i+1}$. For example, when $N=8$, $P_{2}=x_{1}x_{5}x_{3}x_{7}x_{2}x_{6}x_{4}x_{8}$, at this time $x_{7}$ is in the 4th position of $P_{2}$.  
$(1)$ When $N=16$, $x_{7}$ is in the $\boxed{\text{6th}}$ position of $P_{2}$.  
$(2)$ When $N=2^{n}(n\geqslant 8)$, $x_{173}$ is in the $\boxed{3\times2^{n-4}+11}$ position of $P_{4}$.

Present the answer in LaTex format: \boxed{Your answer}
user