
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 $\{1, 2, 3, ..., 23\}$. Define a relation $\succ$ on $S$ such that for $a, b \in S$, $a \succ b$ means either $0 < a - b \le 10$ or $b - a > 12$ under modulo 23 arithmetic. How many ordered triples $(x,y,z)$ from $S$ satisfy $x \succ y$, $y \succ z$, and $z \succ x$?
A) 700
B) 732
C) 759
D) 785
E) 810

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