
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
A right circular cylinder has a height of 10 inches and a base radius of 5 inches. A plane parallel to the base cuts the cylinder into two parts - a smaller cylinder $C$ and a cylindrical frustum $F$. The ratio between the volumes of $C$ and $F$ is equal to $\frac{2}{3}$. Find the height of the smaller cylinder $C$.

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