
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
At the end of a professional bowling tournament, the top 6 bowlers have a playoff. Initially, #6 bowls #5. The loser receives the 6th prize and the winner bowls #4. The loser of this game receives the 5th prize and the winner then bowls #3. The loser of this game receives the 4th prize and the winner bowls #2. The loser of this game receives the 3rd prize, and the winner finally bowls #1. The winner of this final match gets 1st prize and the loser gets 2nd prize. However, there’s a twist: if the winner of the #6 vs #5 match wins all subsequent matches, the bowler who came second in the #6 vs #5 match moves up to receive the 4th prize, pushing others down a rank. How many different orders can the bowlers receive the prizes?

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