
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
Nine barrels. In how many ways can nine barrels be arranged in three tiers so that the numbers written on the barrels that are to the right of any barrel or below it are larger than the number written on the barrel itself? An example of a correct arrangement is having 123 in the top row, 456 in the middle row, and 789 in the bottom row. How many such arrangements are possible?

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