
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
Let $P$ be the pentagon in the coordinate plane with vertices $(0,0), (5,0),$ and $(0,4), (3,3), (-1,3)$. Consider the following transformations of the plane: rotations of $90^{\circ}, 180^{\circ},$ and $270^{\circ}$ counterclockwise around the origin, translation by vector $(5, 0)$, and translation by vector $(0, 4)$. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $P$ to its original position?
A) 6
B) 9
C) 12
D) 15

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