
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
Introduce the relation “ $\rightarrow$ ” between the elements of set $S$ such that:
(1) For any $a, b \in S$, either $a \rightarrow b$ or $b \rightarrow a$, but not both.
(2) For any $a, b, c \in S$, if $a \rightarrow b$ and $b \rightarrow c$, then $c \rightarrow a$.

What is the maximum number of elements that set $S$ can contain?

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