
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 \( f(x)=\frac{a^{x}}{1+a^{x}} \) with \( a>0 \) and \( a \neq 1 \), and \( [m] \) representing the greatest integer less than or equal to a real number \( m \) (i.e., the floor function), find the range of the function \( \left[f(x)-\frac{1}{2}\right] + \left[f(-x)-\frac{1}{2}\right] \).

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