
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
There is a card game called "Twelve Months" that is played only during the Chinese New Year. The rules are as follows:

Step 1: Take a brand new deck of playing cards, remove the two jokers and the four Kings, leaving 48 cards. Shuffle the remaining cards.

Step 2: Lay out the shuffled cards face down into 12 columns, each column consisting of 4 cards.

Step 3: Start by turning over the first card in the first column. If the card is numbered \(N \ (N=1,2, \cdots, 12\), where J and Q correspond to 11 and 12 respectively, regardless of suit, place the card face up at the end of the \(N\)th column.

Step 4: Continue by turning over the first face-down card in the \(N\)th column and follow the same process as in step 3.

Step 5: Repeat this process until you cannot continue. If all 12 columns are fully turned over, it signifies that the next 12 months will be smooth and prosperous. Conversely, if some columns still have face-down cards remaining at the end, it indicates that there will be some difficulties in the corresponding months.

Calculate the probability that all columns are fully turned over.

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