
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 $f(x)=\log_{2}\left(\frac{2}{{x-1}}+a\right)$ is an odd function.  
$(1)$ Find the value of the real number $a$;  
$(2)$ If the inequality with respect to $x$, $2^{f(2^{x})}+3\cdot 2^{x}-b\geqslant 0$, always holds, find the range of real number $b$.

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