
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  $S$  be a subset of the natural numbers such that  $0\in S$ , and for all  $n\in\mathbb N$ , if  $n$  is in  $S$ , then both  $2n+1$  and  $3n+2$  are in  $S$ .  What is the smallest number of elements  $S$  can have in the range  $\{0,1,\ldots, 2019\}$ ?

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