
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 $\{1, 2, 3, \ldots, 29\}$. Define a relation $a \succ b$ among elements $a$, $b$ of $S$ such that $a \succ b$ if either $0 < a - b \leq 14$ or $b - a > 14$. Determine the number of ordered triples $(x, y, z) \in S^3$ such that $x \succ y$, $y \succ z$, and $z \succ x$.

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