
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 \( \{1, 2, 3, ..., 23\} \). For \( a, b \in S \), define \( a \succ b \) to mean that either \( 0 < a - b \le 11 \) or \( b - a > 11 \). 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) 720
B) 759
C) 792
D) 831
E) 860

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