
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
Let $A$ be a finite set of natural numbers. A partition of $A$ into two non-empty disjoint subsets $A_{1}$ and $A_{2}$ is called demonic if the least common multiple of the elements of $A_{1}$ is equal to the greatest common divisor of the elements of $A_{2}$. What is the smallest number of elements that $A$ must have so that there are exactly 2016 demonic partitions?

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