
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 $y=\frac{{b}^{x}+1}{{2}^{x}+a}$, where $(b \gt 0$ and $b\neq 1)$:
- $(1)$ If $a=b=2$, find the range of the function.
- $(2)$ If $a=0$, does there exist a positive number $b$ such that the function is an even function? Explain your reasoning.
- $(3)$ If $a \gt 0$, $b=4$, and the function is strictly increasing on $\left[-1,+\infty \right)$, find the range of real numbers for $a$.

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