
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 $\lfloor x \rfloor$ represent the greatest integer not exceeding $x$, e.g., $\lfloor 1.5 \rfloor = 1$, $\lfloor -1.5 \rfloor = -2$. For the function $f(x) = \frac{a^x}{1 + a^x} \ (a > 0, a \neq 1)$, find the range of values for $g(x) = \lfloor f(x) - \frac{1}{2} \rfloor + \lfloor f(-x) - \frac{1}{2} \rfloor$.

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