
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
Define the function $f$ having the property that for each real number $x$ in its domain, $1/x$ is also in its domain and:
\[
f(x) + f\left(\frac{1}{x}\right) = 1 + x^2.
\]
Moreover, the function satisfies $f(2) = 0$. What is the largest set of real numbers that can be in the domain of $f$?

(a) ${\{x\mid x\ne 0\}}$

(b) ${\{x\mid x<0\}}$

(c) ${\{x\mid x>0\}}$

(d) ${\{x\mid x\ne -1 \ \text{and}\ x\ne 0 \ \text{and}\ x\ne 1\}}$

(e) ${\{-1,1,2\}}$

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