The following table provides data on the different countries and location of their capitals. (the data may not much the actual Latitude, Longitudes) Answer the following questions on the basis of this table. Country Capitals
| No. | Country | Capital | Latitude | Longitude |
|---|---|---|---|---|
| 1 | Argentina | Buenos Aires | \(34.30^\circ S\) | \(58.20^\circ E\) |
| 2 | Australia | Canberra | \(35.15^\circ S\) | \(149.08^\circ E\) |
| 3 | Austria | Vienna | \(48.12^\circ N\) | \(16.22^\circ E\) |
| 4 | Bulgaria | Sofia | \(42.45^\circ N\) | \(23.20^\circ E\) |
| 5 | Brazil | Brasilia | \(15.47^\circ S\) | \(47.55^\circ E\) |
| 6 | Canada | Ottawa | \(45.27^\circ N\) | \(75.42^\circ E\) |
| 7 | Cambodia | Phnom Penh | \(11.33^\circ N\) | \(104.55^\circ E\) |
| 8 | Ecuador | Quito | \(0.15^\circ S\) | \(78.35^\circ E\) |
| 9 | Ghana | Accra | \(5.35^\circ N\) | \(0.6^\circ E\) |
| 10 | Iran | Tehran | \(35.44^\circ N\) | \(51.30^\circ E\) |
| 11 | Ireland | Dublin | \(53.20^\circ N\) | \(6.18^\circ E\) |
| 12 | Libya | Tripoli | \(32.49^\circ N\) | \(13.07^\circ E\) |
| 13 | Malaysia | Kuala Lumpur | \(3.9^\circ N\) | \(101.41^\circ E\) |
| 14 | Peru | Lima | \(12.05^\circ S\) | \(77.0^\circ E\) |
| 15 | Poland | Warsaw | \(52.13^\circ N\) | \(21.0^\circ E\) |
| 16 | New Zealand | Wellington | \(41.17^\circ S\) | \(174.47^\circ E\) |
| 17 | Saudi Arabia | Riyadh | \(24.41^\circ N\) | \(46.42^\circ E\) |
| 18 | Spain | Madrid | \(40.25^\circ N\) | \(3.45^\circ W\) |
| 19 | Sri Lanka | Colombo | \(6.56^\circ N\) | \(79.58^\circ E\) |
| 20 | Zambia | Lusaka | \(15.28^\circ S\) | \(28.16^\circ E\) |
A train travels from Station A to Station E, passing through stations B, C, and D, in that order. The train has a seating capacity of 200. A ticket may be booked from any station to any other station ahead on the route, but not to any earlier station. A ticket from one station to another reserves one seat on every intermediate segment of the route. For example, a ticket from B to E reserves a seat in the intermediate segments B– C, C– D, and D–E. The occupancy factor for a segment is the total number of seats reserved in the segment as a percentage of the seating capacity. The total number of seats reserved for any segment cannot exceed 200. The following information is known. 1. Segment C– D had an occupancy factor of 952. Exactly 40 tickets were booked from B to C and 30 tickets were booked from B to E. 3. Among the seats reserved on segment D– E, exactly four-sevenths were from stations before C. 4. The number of tickets booked from A to C was equal to that booked from A to E, and it was higher than that from B to E. 5. No tickets were booked from A to B, from B to D and from D to E. 6. The number of tickets booked for any segment was a multiple of 10.
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: