
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 square $WXYZ$ in the coordinate plane has vertices at the points $W(2,2), X(-2,2), Y(-2,-2), Z(2,-2)$. Consider these transformations:
$\quad \bullet \quad$ $R_1$, a rotation of $180^\circ$ around the origin;
$\quad \bullet \quad$ $R_2$, a rotation of $90^\circ$ clockwise around the origin;
$\quad \bullet \quad$ $M_X$, a reflection across the $x$-axis; and
$\quad \bullet \quad$ $M_Y$, a reflection across the $y$-axis.
Find the number of valid sequences of 10 transformations, selected from $\{R_1, R_2, M_X, M_Y\}$, that return all labeled vertices to their original positions.

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