
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 square $PQRS$ with vertices at the points $P(2,2), Q(-2,2), R(-2,-2), S(2,-2)$ placed in the coordinate plane. Let's define a new set of transformations:
- $A$, a rotation of $45^{\circ}$ counterclockwise around the origin;
- $B$, a rotation of $45^{\circ}$ clockwise around the origin;
- $M$, reflection across the line $y=x$;
- $N$, reflection across the line $y=-x$.

Determine the number of sequences of $10$ transformations chosen from $\{A, B, M, N\}$ that will return all the labeled vertices back to their original positions.
A) 35
B) 56
C) 70
D) 84
E) 210

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