
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
In the game of Guess the Card, two players each have a  $\frac{1}{2}$  chance of winning and there is exactly one winner. Sixteen competitors stand in a circle, numbered  $1,2,\dots,16$  clockwise. They participate in an  $4$ -round single-elimination tournament of Guess the Card. Each round, the referee randomly chooses one of the remaining players, and the players pair off going clockwise, starting from the chosen one; each pair then plays Guess the Card and the losers leave the circle. If the probability that players  $1$  and  $9$  face each other in the last round is  $\frac{m}{n}$  where  $m,n$  are positive integers, find  $100m+n$ .

*Proposed by Evan Chen*

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