
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 a triangle in the coordinate plane with vertices at $(0,0)$, $(6,0)$, and $(0,4)$. The five transformations are the same as in the original problem: rotations of $90^{\circ}, 180^{\circ},$ and $270^{\circ}$ counterclockwise around the origin, and reflections across the $x$-axis and the $y$-axis. Determine how many sequences of three of these transformations (not necessarily distinct) will return $T$ to its exact original position.
A) 5
B) 6
C) 7
D) 8
E) 9

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