
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
A square $WXYZ$ in the coordinate plane has corners at $W(2,3), X(3,2), Y(2,1),$ and $Z(1,2)$. Consider these four transformations:
• $A,$ a rotation of $90^{\circ}$ clockwise around point $(2,2)$;
• $B,$ a rotation of $90^{\circ}$ counterclockwise around point $(2,2)$;
• $C,$ a reflection across the line $y = 2$; and
• $D,$ a reflection across the line $x = 2$.
Determine the number of sequences of $12$ transformations, chosen from $\{A, B, C, D\}$, that will return all the labeled vertices to their original positions.

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