
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 the function $f(x)$ satisfies $2f\left(x+2\right)=f\left(x\right)$, when $x\in \left(0,2\right)$, $f\left(x\right)=\ln x+ax\left(a < - \frac{1}{2}\right)$, and the maximum value of the function when $x\in \left(-4,-2\right)$ is $-4$.

(I) Find the analytic expression of the function $f(x)$ when $x\in \left(0,2\right)$;

(II) Determine whether there exists a real number $b$ such that the inequality $\frac{x-b}{f\left(x\right)+x} > \sqrt{x}$ holds for $x\in \left(0,1\right)\cup \left(1,2\right)$. If such a $b$ exists, find its range of values; if not, explain why.

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