
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 \( f(x) \) with a domain of \((0, +\infty)\) such that for any \( x \in (0, +\infty) \), it always holds that
\[ f(2x) = 2f(x); \]
and for \( x \in (1, 2] \), \( f(x) = 2 - x \).

The following conclusions are presented:
1. For any \( m \in \mathbf{Z} \), \( f(2^m) = 0 \);
2. The range of the function \( f(x) \) is \([0, +\infty)\);
3. There exists an \( n \in \mathbf{Z} \) such that \( f(2^n + 1) = 9 \);
4. The necessary and sufficient condition for "the function \( f(x) \) is monotonically decreasing on the interval \((a, b)\)" is "there exists a \( k \in \mathbf{Z} \) such that \((a, b) \subseteq (2^k, 2^{k+1}) \).

Identify the numbers of all correct conclusions.

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