
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 that the domain of the function $f(x)=\left( \frac{1}{2} \right)^{\sqrt{-x^2-2x}}$ is the set $A$, and its range is the set $B$.
$(1)$ Find the sets $A$ and $B$;
$(2)$ Let the function $g(x)=k+\log_{2}x$ where $x \in B$. If the range of the function $g(x)$ is a proper subset of set $A$, find the range of the real number $k$.

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