
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
Arrange the numbers $1, 2, 3, 4, 6$ in a circle, and determine how many unique "bad" arrangements exist if an arrangement is considered “bad” when there is at least one number between $1$ and $16$ that cannot be the sum of a subset of consecutive numbers in the circle.

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