
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 $A$ consisting of all functions $f(x)$ that simultaneously satisfy the following two properties:
1. The function $f(x)$ is a monotonic function within 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 $f(x) = x^3$ belongs to $M$. If yes, find all intervals $[a, b]$ that satisfy property 2. If no, explain why.
   - (2) Determine whether there exists a real number $t$ such that $h(x) = \sqrt{x-1} + t \in M$. If such $t$ exists, find the range of $t$. If not, explain why.

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