
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 triangle $PQR$ in the coordinate plane with vertices at $P(1,2)$, $Q(-1,2)$, and $R(0,-1)$. We define four transformations:
- $L$, a rotation of $120^\circ$ counterclockwise around the centroid of the triangle;
- $R$, a rotation of $120^\circ$ clockwise around the centroid of the triangle;
- $H$, a reflection across the line connecting $P$ and $Q$;
- $V$, a reflection across the line $y = -1$ (horizontal line passing through $R$).

Each transformation maps the triangle onto itself, but with vertices changing their positions. How many sequences of $15$ transformations selected from $\{L, R, H, V\}$ will return all labeled vertices to their original positions?
- **A)** $28$
- **B)** $35$
- **C)** $56$
- **D)** $70$
- **E)** $84$

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