
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
Square $EFGH$ in the coordinate plane has vertices at the points $E(2,2), F(-2,2), G(-2,-2), H(2,-2)$. Consider these transformations:
- $L$, a rotation of $90^{\circ}$ counterclockwise around the origin;
- $R$, a rotation of $90^{\circ}$ clockwise around the origin;
- $H$, a reflection across the $x$-axis;
- $V$, a reflection across the $y$-axis;
- $S$, a scaling by a factor of 2 centered at the origin.
How many sequences of $24$ transformations chosen from $\{L, R, H, V, S\}$ will send all of the labeled vertices back to their original positions?
A) $286$
B) $300$
C) $400$
D) $455$
E) $500$

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