
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 $\{1,2,3,...,20\}$. For $a,b \in S$, define $a \succ b$ to mean that either $0 < a - b \le 10$ or $b - a > 10$. How many ordered quadruples $(w, x,y,z)$ of elements of $S$ have the property that $w \succ x$, $x \succ y$, $y \succ z$, and $z \succ w$?
A) 154000
B) 156000
C) 157120
D) 158000
E) 159000

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