
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
Consider a regular hexagon $ABCDEF$ in the coordinate plane with vertices placed symmetrically around the origin. The transformations are:
- $T_1$: a rotation of $60^{\circ}$ clockwise around the origin
- $T_2$: a rotation of $60^{\circ}$ counterclockwise around the origin
- $T_3$: a reflection across the x-axis
- $T_4$: a reflection across the y-axis
How many sequences of $18$ transformations chosen from $\{T_1, T_2, T_3, T_4\}$ will send all of the labeled vertices back to their original positions?
A) 280
B) 285
C) 286
D) 287
E) 288

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