
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 $\{1, 2, 3, \ldots, 24\}$. For $a, b \in S$, define $a \succ b$ to mean either $0 < a - b \le 12$ or $b - a > 12$. Determine how many ordered triples $(x,y,z)$ of elements of $S$ have the property that $x \succ y$, $y \succ z$, and $z \succ x$.
A) 1020
B) 1080
C) 1100
D) 1150
E) 1200

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