
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 $(0, 0)$, $(5, 0)$, $(5, 2)$, and $(0, 2)$. The set of transformations available includes rotations of $90^\circ, 180^\circ,$ and $270^\circ$ counterclockwise around the origin, reflection across the $x$-axis, reflection across the $y$-axis, and a translation of $(2, 3)$. Determine how many of the $6^3 = 216$ sequences of three of these transformations (not necessarily distinct) will return $R$ back to its original position.
A) 12
B) 15
C) 17
D) 20
E) 25

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