
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
Six bowlers participated in a professional bowling tournament. In round one, bowlers #4, #5, and #6 compete in a sequence of two matches: first #6 bowls #5, the winner bowls #4. The top two among these three then advance, with the loser of the second game taking the 6th prize, and the second last of the first match taking 5th prize. The next rounds follow the playoff format with a slight change: the final winner of the initial round bowls against #3, and continue as previously described with the loser of each game receiving the next available descending prize, and the final match winner and loser receiving 1st and 2nd prizes respectively. In how many orders can bowlers #1 through #6 receive the prizes?

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