
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
A regular hexagon $ABCDEF$ in the coordinate plane has vertices at the points $A(1,\sqrt{3}), B(2,0), C(1,-\sqrt{3}), D(-1,-\sqrt{3}), E(-2,0),$ and $F(-1,\sqrt{3})$. Analyze the following four transformations:
- $R_a,$ a rotation of $60^{\circ}$ clockwise around the origin;
- $R_b,$ a rotation of $120^{\circ}$ clockwise around the origin;
- $H,$ a reflection across the $x$-axis; and
- $V,$ a reflection across the y-axis.
Each transformation maps the hexagon onto itself albeit with changes in the vertex labeling. How many sequences of $24$ transformations, chosen from $\{R_a, R_b, H, V\}$ will send all of the labeled vertices back to their original positions?
A) $4^{23}$
B) $4^{24}$
C) $3^{24}$
D) $2^{48}$
E) $6^{12}$

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