
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 $f(x)=\dfrac{b-2^{x}}{2^{x+1}+a}$ is an odd function defined on $\mathbb{R}$.

$(1)$ Find the values of real numbers $a$ and $b$;

$(2)$ Determine the monotonicity of $f(x)$ on $\mathbb{R}$;

$(3)$ If $f(k\cdot 3^{x})+f(3^{x}-9^{x}+2) > 0$ holds for any $x \geqslant 1$, find the range of the real number $k$.

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