One day, two other traders, Dane and Emily, joined Abdul, Bikram, and Chetan for trading in the shares of XYZ Ltd.
Dane’s strategy: Buy equal numbers of shares at 10 am, 11 am, and 12 noon, and sell the same numbers at 1 pm, 2 pm, and 3 pm.
Emily’s strategy: Buy shares using all her money at 10 am and sell all of them at 12 noon; then buy shares again using all her money at 1 pm and sell all of them at the close of the day at 3 pm.
At the close of the day, the following was observed:





For any natural number $k$, let $a_k = 3^k$. The smallest natural number $m$ for which \[ (a_1)^1 \times (a_2)^2 \times \dots \times (a_{20})^{20} \;<\; a_{21} \times a_{22} \times \dots \times a_{20+m} \] is: