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$10$ players – $\text{P1, P2,} \dots, \text{P10}$ – competed in an international javelin throw event. The number (after $\text{P}$) of a player reflects his rank at the beginning of the event, with rank $1$ going to the topmost player. There were two phases in the event with the first phase consisting of rounds $1, 2,$ and $3,$ and the second phase consisting of rounds $4, 5,$ and $6.$ A throw is measured in terms of the distance it covers (in meters, up to one decimal point accuracy), only if the throw is a ‘valid’ one. For an invalid throw, the distance is taken as zero. A player’s score at the end of a round is the maximum distance of all his throws up to that round. Players are re-ranked after every round based on their current scores. In case of a tie in scores, the player with a prevailing higher rank retains the higher rank. This ranking determines the order in which the players go for their throws in the next round.

In each of the rounds in the first phase, the players throw in increasing order of their latest rank, i.e. the player ranked $1$ at that point throws first, followed by the player ranked $2$ at that point and so on. The top six players at the end of the first phase qualify for the second phase. In each of the rounds in the second phase, the players throw in decreasing order of their latest rank i.e. the player ranked $6$ at that point throws first, followed by the player ranked $5$ at that point and so on. The players ranked $1, 2,$ and $3$ at the end of the sixth round receive gold, silver, and bronze medals respectively. 

All the valid throws of the event were of distinct distances (as per stated measurement accuracy). The tables below show distances (in meters) covered by all valid throws in the first and the third round in the event.

Distances covered by all the valid throws in the first round

$$\begin{array}{|c|c|c|} \hline \textbf{Player} & \textbf{Distance (in m)} \\\hline \text{P1} & 82.9 \\\hline \text{P3} & 81.5 \\\hline \text{P5} & 86.4 \\\hline \text{P6} & 82.5 \\\hline \text{P7} & 87.2 \\\hline \text{P9} & 84.1 \\\hline \end{array}$$

Distances covered by all the valid throws in the third round

$$\begin{array}{|c|c|c|} \hline \textbf{Player} & \textbf{Distance (in m)} \\\hline \text{P1} & 88.6 \\\hline \text{P3} & 79.0 \\\hline \text{P9} & 81.4 \\\hline \end{array}$$

The following facts are also known.

  1. Among the throws in the second round, only the last two were valid. Both the throws enabled these players to qualify for the second phase, with one of them qualifying with the least score. None of these players won any medal.
  2. If a player throws first in a round $\text{AND}$ he was also the last (among the players in the current round) to throw in the previous round, then the player is said to get a double. Two players got a double.
  3. In each round of the second phase, exactly one player improved his score. Each of these improvements was by the same amount.
  4. The gold and bronze medalists improved their scores in the fifth and sixth rounds respectively. One medal winner improved his score in the fourth round.
  5. The difference between the final scores of the gold medalist and the silver medalist, as well as the difference between the final scores of the silver medalist and the bronze medalist was $1.0 \; \text{m}.$

By how much did the gold medalist improve his score (in $m$) in the second phase?

  1. $1.2$
  2. $1.0$
  3. $2.4$
  4. $2.0$
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