Mulayam Software Co., before selling a package to its clients, follows the given schedule.
| Month | Stage | Cost (Rs. $'000$ per man / month) |
|---|---|---|
| 1–2 | Specification | $40$ |
| 3–4 | Design | $20$ |
| 5–8 | Coding | $10$ |
| 9–10 | Testing | $15$ |
| 11–15 | Maintenance | $10$ |
The number of people employed in each month is:
| Month | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ | $13$ | $14$ | $15$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Number of people employed | $2$ | $3$ | $4$ | $3$ | $4$ | $5$ | $5$ | $4$ | $4$ | $1$ | $3$ | $3$ | $1$ | $1$ | $1$ |




| 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: