
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
Given $T' = \{(0,0),(5,0),(0,4)\}$ be a triangle in the coordinate plane, determine the number of sequences of three transformations—comprising rotations by $90^{\circ}, 180^{\circ},$ and $270^{\circ}$ counterclockwise around the origin, and reflections across the lines $y = x$ and $y = -x$—that will map $T'$ onto itself.

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