
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
There are  $256$  players in a tennis tournament who are ranked from  $1$  to  $256$ , with  $1$  corresponding to the highest rank and  $256$  corresponding to the lowest rank. When two players play a match in the tournament, the player whose rank is higher wins the match with probability  $\frac{3}{5}$ .

In each round of the tournament, the player with the highest rank plays against the player with the second highest rank, the player with the third highest rank plays against the player with the fourth highest rank, and so on. At the end of the round, the players who win proceed to the next round and the players who lose exit the tournament. After eight rounds, there is one player remaining and they are declared the winner.

Determine the expected value of the rank of the winner.

Present the answer in LaTex format: \boxed{Your answer}
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