
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 league, the bottom two of the top seven bowlers (#6 and #7) have a playoff match, and the loser receives $7$th position. The winner of this match faces #5 in another game. The loser is awarded the $6$th position, and the winner continues to face #4, #3, #2, and #1 consecutively in separate matches. Similar to the original structure, each game's loser receives the prize one rank better than the previous game's position, leading up to the match with #1 where the winner gets the 1st prize and the loser gets the 2nd prize. Calculate the number of different orders in which bowlers #1 through #7 could be awarded prizes.

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