The amount of money invested (rupees in crores) in the core infrastructure areas of two districts, Chittoor and Khammam of Andhra Pradesh, is as follows.
| Chittoor district | Khammam district | ||||
|---|---|---|---|---|---|
| Core area | $1995$ | $1996$ | Core area | $1995$ | $1996$ |
| Electricity | $815.2$ | $1054.2$ | Electricity | $2065.8$ | $2365.1$ |
| Chemical | $389.5$ | $476.7$ | Chemical | $745.3$ | $986.4$ |
| Thermal | $632.4$ | $565.9$ | Thermal | $1232.7$ | $1026.3$ |
| Solar | $468.1$ | $589.6$ | Solar | $1363.5$ | $1792.1$ |
| Nuclear | $617.9$ | $803.1$ | Nuclear | $1674.3$ | $2182.1$ |
| Total | $2923.1$ | $3489.5$ | Total | $7081.6$ | $8352.0$ |




| 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: