
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 set M that consists of all functions $f(x)$ with the following property: within the domain $D$, there exists an $x_0$ such that $f(x_0+1) = f(x_0) + f(1)$.  
(1) Does the function $f(x) = \frac{1}{x}$ belong to set M? Justify your reason;  
(2) If the function $f(x) = k \cdot 2^x + b$ belongs to set M, find the conditions that real numbers $k$ and $b$ must satisfy;  
(3) Assume the function $f(x) = \lg \left( \frac{a}{x^2+2} \right)$ belongs to set M, find the range of the real number $a$.

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