
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 function $f(x)=\frac{1}{{2}^{x}+a}-\frac{1}{2}$ with domain $\mathbb{R}$ is an odd function,  
$(1)$ Determine the monotonicity of the function $f(x)$ (no proof required), and find the range of $f(x)$;  

$(2)$ If for any $x \in [-1,1]$, the inequality $f(k-{x}^{2})+f(2-x) > 0$ always holds, find the range of $k$.

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