
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 that $S$ is the set $\{1,2,3,...,19\}$, for $a,b \in S$, define $a \succ b$ to mean that either $0 < a - b \le 9$ or $b - a > 9$. Calculate the number of ordered triples $(x,y,z)$ of elements of $S$ such that $x \succ y$, $y \succ z$, and $z \succ x$.

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