
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 chairs in a row are to be occupied by six students and Professors Alpha, Beta, and Gamma. Professors arrive first and are to choose their seats so that they are each surrounded by students, although they can now also choose the end chairs if followed or preceded by a student. Calculate the number of ways Professors Alpha, Beta, and Gamma can choose their chairs.

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