
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
A  $3n$ -table is a table with three rows and  $n$  columns containing all the numbers  $1, 2, …, 3n$ . Such a table is called *tidy* if the  $n$  numbers in the first row appear in ascending order from left to right, and the three numbers in each column appear in ascending order from top to bottom. How many tidy  $3n$ -tables exist?

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