
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 $T$ be the triangle in the coordinate plane with vertices at $(1,1), (5,1),$ and $(1,4)$. Consider the following isometries of the plane: rotations of $60^\circ$, $120^\circ$, and $180^\circ$ counterclockwise around the point $(1,1)$, reflection across the $x$-axis, and reflection across the $y$-axis. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position?
A) 10
B) 12
C) 22
D) 24
E) 15

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