
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
Given the set $S$ of integers from $1$ to $19$, define $a \succ b$ to mean that either $0 < a - b \le 8$ or $b - a > 10$. Determine the number of ordered triples $(x,y,z)$ of elements from $S$ that satisfy the conditions $x \succ y$, $y \succ z$, and $z \succ x$.

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