
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 the function \( f(x)=\frac{x^{3}}{1+x^{3}} \), find the value of the sum

\[ 
f\left(\frac{1}{1}\right)+f\left(\frac{2}{1}\right)+\ldots+f\left(\frac{2007}{1}\right) + f\left(\frac{1}{2}\right)+f\left(\frac{2}{2}\right)+\ldots+f\left(\frac{2007}{2}\right) + \ldots + f\left(\frac{1}{2007}\right)+f\left(\frac{2}{2007}\right)+\ldots+f\left(\frac{2007}{2007}\right)
\]

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