
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 a function $f(x)$ satisfying $2f(x+2) = f(x)$, when $x \in (0, 2)$, $f(x) = \ln x + ax$ $(a < -\frac{1}{2})$, and when $x \in (-4, -2)$, the maximum value of $f(x)$ is -4.  
(1) Find the analytic expression of the function $f(x)$ when $x \in (0, 2)$;  
(2) Determine whether there exists a real number $b$ such that the inequality $\frac{x - b}{f(x) + x} > \sqrt{x}$ holds for all $x \in (0, 1) \cup (1, 2)$. If it exists, find the set of values of the real number $b$; if it does not exist, explain why.

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