
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
A ladder style tournament is held with  $2016$  participants. The players begin seeded  $1,2,\cdots 2016$ . Each round, the lowest remaining seeded player plays the second lowest remaining seeded player, and the loser of the game gets eliminated from the tournament. After  $2015$  rounds, one player remains who wins the tournament. If each player has probability of  $\tfrac{1}{2}$  to win any game, then the probability that the winner of the tournament began with an even seed can be expressed has  $\tfrac{p}{q}$  for coprime positive integers  $p$  and  $q$ . Find the remainder when  $p$  is divided by  $1000$ .

[i]Proposed by Nathan Ramesh

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