
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
Consider a square $WXYZ$ in the coordinate plane with vertices at $W(2, 2)$, $X(-2, 2)$, $Y(-2, -2)$, and $Z(2, -2)$. We are given four transformations:
- $A$: $90^{\circ}$ counterclockwise rotation around the origin;
- $B$: $90^{\circ}$ clockwise rotation around the origin;
- $C$: reflection across the $x$-axis;
- $D$: reflection across the $y$-axis.
Each of these transformations maps the square onto itself but alters the positions of the labeled vertices. Determine how many sequences of 12 transformations using $\{A, B, C, D\}$ will restore all labeled vertices to their original positions.
A) 207900
B) 311850
C) 415800
D) 623700
E) 830400

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