
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 $f(x)=\log_{2}[1+2^{x}+a\cdot(4^{x}+1)]$  
$(1)$ When $a=-1$, find the domain of the function $f(x)$;  
$(2)$ When $a=-\frac{1}{2}$, the graph of the function $h(x)=f(x)-x$ has no intersection with $y=b(0\leqslant x\leqslant 1)$, find the range of the real number $b$.

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