
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
The function $f$ has the property that for each real number $x$ in its domain, $2x$ is also in its domain and 
\[f(x) + f\left(2x\right) = x + 1.\]
What is the largest set of real numbers that can be in the domain of $f$?

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

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

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

(d) $\{x\mid x \neq -\frac{1}{2}\ \text{and}\ x \neq 0\}$

(e) $\{x\mid x = -\frac{1}{2}, x = 0\}$

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