
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 $f(x) = \frac{x^2}{1+x^2}$.

$(1)$ Calculate the values of $f(2) + f\left(\frac{1}{2}\right)$, $f(3) + f\left(\frac{1}{3}\right)$, $f(4) + f\left(\frac{1}{4}\right)$, and infer a general conclusion (proof not required);

$(2)$ Calculate the value of $2f(2) + 2f(3) + \ldots + 2f(2017) + f\left(\frac{1}{2}\right) + f\left(\frac{1}{3}\right) + \ldots + f\left(\frac{1}{2017}\right) + \frac{1}{2^2}f(2) + \frac{1}{3^2}f(3) + \ldots + \frac{1}{2017^2} \cdot f(2017)$.

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