
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 numbers $1, 2, 3, 4, 6$ are to be arranged in a circle. An arrangement is considered $\textit{bad}$ if it is not true that for every $n$ from $1$ to $16$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Determine the number of different bad arrangements.

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