Let $a_1, a_2, \ldots$ be a sequence with the following properties.

(i) $a_1 = 1$, and

(ii) $a_{2n}=n\cdot a_n$ for any positive integer $n$.

What is the value of $a_{2^{100}}$?
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $a_1, a_2, \ldots$ be a sequence with the following properties.

(i) $a_1 = 1$, and

(ii) $a_{2n}=n\cdot a_n$ for any positive integer $n$.

What is the value of $a_{2^{100}}$?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
2^{4950}
Let $a_1, a_2, \ldots$ be a sequence with the following properties.

(i) $a_1 = 1$, and

(ii) $a_{2n}=n\cdot a_n$ for any positive integer $n$.

What is the value of $a_{2^{100}}$?
train