
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 $M$ consisting of all functions $f(x)$ that simultaneously satisfy the following two properties:  
① The function $f(x)$ is a monotonic function on its domain.  
② There exists an interval $[a, b]$ within the domain of $f(x)$ such that the range of $f(x)$ on $[a, b]$ is $[\frac{a}{2}, \frac{b}{2}]$.  
(1) Determine whether the function $f(x)=x+ \frac{2}{x} (x>0)$ belongs to $M$, and explain the reason.  
(2) Determine whether $g(x)=-x^3$ belongs to $M$, and explain the reason. If it does, find the interval $[a, b]$ that satisfies property ②.

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