
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 $M$ is the collection of all functions $f(x)$ that simultaneously satisfy the following two properties:

$\quad\quad①$ The function $f(x)$ is monotonic on its domain.

$\quad\quad②$ There exists an interval $[a, b]$ in 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 $②$.

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