
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 $f(x) = \log_{2}[1 + 2^{x} + a(4^{x} + 1)]$,

(1) When $a = -1$, find the domain of the function $f(x)$;

(2) When $x \in (-\infty, 1]$, the function $f(x)$ is meaningful, find the range of the real number $a$;

(3) When $a = -\frac{1}{2}$, the graph of the function $y = f(x)$ has no intersection with $y = x + b$ ($0 \leq x \leq 1$), find the range of the real number $b$.

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