
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 dilation of the plane that sends a square with vertices initially at $S(3,3)$, $T(7,3)$, $U(7,7)$, and $V(3,7)$ to a larger square with vertices $S'(6,6)$, $T'(12,6)$, $U'(12,12)$, and $V'(6,12)$. Calculate the distance the origin $(0,0)$ moves under this transformation.

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