
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 triangle \( T \) have vertices at \( (0,0) \), \( (6,0) \), and \( (0,4) \) in the coordinate plane. Consider these transformations: a \( 180^\circ \) counterclockwise rotation about the origin, reflection across the \( x \)-axis, reflection across the \( y \)-axis, and a dilation by a factor of \( 2 \) centered at the origin. Determine how many of the \( 256 \) sequences of four transformations (not necessarily distinct) will return \( T \) to its original position.
A) 11
B) 12
C) 13
D) 14
E) 15

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