
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
Given two cones, cone A and cone B, with equal base radii and side areas $S_{A}$ and $S_{B}$, and volumes $V_{A}$ and $V_{B}$, respectively, determine the ratio of the volumes $\frac{V_{A}}{{V}_{B}}$ when the lateral surface of cone A can be unfolded into a semicircle, given that $\frac{{S}_{A}}{{S}_{B}}=\frac{2}{3}$.

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