
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 that set $A$ is the collection of all functions $f(x)$ that satisfy the condition: there exists a real number ${{x}_{0}}$ in the domain such that $f({{x}_{0}}+1)+f({{x}_{0}})=f(1)$.

(I) Determine whether the power function $f(x)={{x}^{-1}}$ belongs to set $A$ and explain the reasoning.

(II) Let $g(x)= \lg \frac{{{2}^{x}}+a}{b}$, $x\in (-\infty ,2)$,

(i) When $b=1$, if $g(x)\in A$, find the range of values for $a$.

(ii) If for any $a\in (0,2)$, $g(x)\in A$, find the range of values for $b$.

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