
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
In the full permutation of 4 $x$, 3 $y$, and 2 $z$, determine the number of permutations where the patterns $x x x x$, $y y y$, and $z z$ do not appear. Let $A_{1}$ represent the set of permutations of 9 characters where the pattern $x x x x$ appears, $A_{2}$ represent the set where the pattern $y y y$ appears, and $A_{3}$ represent the set where the pattern $z z$ appears.

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