
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 $a > 0$, consider the function $f(x)= \frac {1}{1+a\cdot 2^{x}}$
$(1)$ If $a=1$, determine the inverse function $f^{-1}(x)$ of $f(x)$.
$(2)$ Find the maximum value of the function $y=f(x)\cdot f(-x)$ (expressed in terms of $a$).
$(3)$ Let $g(x)=f(x)-f(x-1)$. If for all $x \in (-\infty, 0]$, $g(x) \geq g(0)$ always holds, determine the range of values for $a$.

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