
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
Let $ABC$ be a triangle with centroid $G$. Suppose that vertices $A$, $B$, and $C$ of the triangle are rotated around $G$ by an angle $\theta = 120^\circ$ and scaled by a factor $s$, resulting in a new triangle $A'B'C'$ such that vertex $A'$, $B'$, and $C'$ are positioned by multiplying the vector from $G$ to $A$, $B$, and $C$ by $s$ and rotating by $\theta$. Determine the value of $s$ such that triangle $A'B'C'$ is similar to triangle $ABC$.

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