
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 rectangle $R$ in the coordinate plane with vertices at $(0,0), (6,0), (6,4), (0,4)$. You have the following transformations: rotation of $180^\circ$ clockwise around the origin and reflections across the $x$-axis, $y$-axis, and the line $y=x$. Determine how many of the $64$ sequences of three transformations (not necessarily distinct) will return $R$ to exactly overlap its original position.
A) 12
B) 18
C) 24
D) 27
E) 30

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