
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
Rectangle $ABCD$ in the coordinate plane has vertices at the points $A(2,1), B(-2,1), C(-2,-1),$ and $D(2,-1)$. Consider the following three transformations:
$\quad \bullet \quad$ $L$, a rotation of $180^\circ$ around the origin;
$\quad \bullet \quad$ $H$, a reflection across the $x$-axis;
$\quad \bullet \quad$ $V$, a reflection across the $y$-axis.
Each of these transformations maps the rectangle onto itself, but the positions of the labeled vertices will change. Determine how many sequences of $10$ transformations chosen from $\{L, H, V\}$ will send all of the labeled vertices back to their original positions.
A) 15
B) 18
C) 21
D) 28
E) 35

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