Answer the questions based on the following information, which gives data about certain coffee producers in India.
| Production ($'000$ tonnes) | Capacity utilisation ($\%$) | Sales ($'000$ tonnes) | |
|---|---|---|---|
| Brooke Bond | $2.97$ | $76.50$ | $2.55$ |
| Nestle | $2.48$ | $71.20$ | $2.03$ |
| Lipton | $1.64$ | $64.80$ | $1.26$ |
| MAC | $1.54$ | $59.35$ | $1.47$ |
| Total (including others) | $11.60$ | $61.30$ | $10.67$ |




| A | B | C | D | Average |
|---|---|---|---|---|
| 3 | 4 | 4 | ? | 4 |
| 3 | ? | 5 | ? | 4 |
| ? | 3 | 3 | ? | 4 |
| ? | ? | ? | ? | 4.25 |
| 4 | 4 | 4 | 4.25 |
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: