
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 \( f(x) = \frac{a^x}{1 + a^x} \) where \( a > 0 \) and \( a \neq 1 \). Let \([m]\) denote the greatest integer less than or equal to the real number \( m \). Find the range of the function \( \left\lfloor f(x) - \frac{1}{2} \right\rfloor + \left\lfloor f(-x) - \frac{1}{2} \right\rfloor \).

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