Question 24Let $a_1,$ $a_2,$ $\ldots$ be a sequence with the following properties: $a_1 = 1,$ and $a_{2n} = n \cdot a_n$ for any positive integer $n.$ What is the value of $a_{2^{100}}?$$1$$2^{99}$$2^{100}$$2^{4950}$$2^{9999}$