
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 $(0,0), (6,0),$ and $(0,4)$. Consider the following isometries of the plane: rotations of $90^\circ$ and $180^\circ$ counterclockwise around the origin, and reflections across the $x$-axis, $y$-axis, and the line $y=x$. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position?
A) 12
B) 15
C) 17
D) 18
E) 20

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