
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 $f(x)=\frac{x^2}{1+x^2}$.
(I) Find the values of $f(2)+f(\frac{1}{2})$, $f(3)+f(\frac{1}{3})$, and $f(4)+f(\frac{1}{4})$;
(II) Make a general conjecture and provide a proof;
(III) Evaluate $f(1)+f(2)+\dots+f(2011)+f(\frac{1}{2011})+f(\frac{1}{2010})+\dots+f(\frac{1}{2})+f(1)$.

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