
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 function $f(x)=\frac{-{2}^{x}+b}{{2}^{x+1}+a}$ defined on $R$ is an odd function.<br/>$(1)$ Find the values of the real numbers $a$ and $b$;<br/>$(2)$ Determine the monotonicity of the function $f\left(x\right)$;<br/>$(3)$ If the inequality $f\left(k\right)+f(\cos ^{2}\theta -2\sin \theta )\leqslant 0$ has a solution for any $θ∈(-\frac{π}{2},\frac{π}{2})$, find the range of real number $k$.

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