
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
Let $f(x) = \frac{x^2}{1+x^2}$.
(1) Find the values of $f(2)+f\left(\frac{1}{2}\right)$, $f(3)+f\left(\frac{1}{3}\right)$, and $f(4)+f\left(\frac{1}{4}\right)$, respectively;
(2) Formulate a general conjecture based on the results and provide a proof;
(3) Evaluate $f(1) + f(2) + f(3) + \ldots + f(2015) + f\left(\frac{1}{2}\right) + f\left(\frac{1}{3}\right) + \ldots + f\left(\frac{1}{2015}\right)$.

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