
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 $ABCD$ has side length 4 units. It is first rotated 180 degrees clockwise about vertex $A$. Subsequently, it is reflected across the diagonal $BD$. Determine the total path length traveled by vertex $C$ through both transformations.
A) $8\sqrt{2}$ units
B) $4\sqrt{2} + 4$ units
C) $8 + 4\sqrt{2}$ units
D) $16$ units

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