
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 $(0,0)$, $(5,0)$, and $(0,4)$. Consider the following six isometries (rigid transformations) of the plane: rotations of $60^{\circ}$, $120^{\circ}$, and $180^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, reflection across the $y$-axis, and a scaling transformation by a factor of $1$ (identity scaling). How many of the $216$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position?
A) 15  
B) 17   
C) 20  
D) 22   
E) 25

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