
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
For any natural number \( n \), find the value of the expression

\[ 
\sqrt[3]{\frac{1 \cdot 2 \cdot 4+2 \cdot 4 \cdot 8+\ldots+n \cdot 2n \cdot 4n}{1 \cdot 3 \cdot 9+2 \cdot 6 \cdot 18+\ldots+n \cdot 3n \cdot 9n}}
\]

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