
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
The Annual Intergalactic Math Olympiad (AIMO) is organized by a committee of four Jupitarians, four Saturnians, and four Neptunians. They sit at a round table with chairs numbered from $1$ to $12$ in clockwise order. According to the committee rules, a Jupitarian must occupy chair $1$ and a Neptunian must occupy chair $12$. Furthermore, no Jupitarian can sit immediately to the left of a Neptunian, no Neptunian can sit immediately to the left of a Saturnian, and no Saturnian can sit immediately to the left of a Jupitarian. The number of possible seating arrangements for the committee is \( M \cdot (4!)^3 \). Find \( M \).

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