
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 the set $\{1,2,3,...,20\}$. Define a relation $a \succ b$ for $a, b \in S$ where $a \succ b$ means either $0 < a - b \le 10$ or $b - a > 10$. Determine the number of ordered triples $(x, y, z)$ in $S$ such that $x \succ y$, $y \succ z$, and $z \succ x$.
A) 700
B) 720
C) 740
D) 760
E) 780

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