
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 the domain \( (0, +\infty) \) which satisfies:
1. For any \( x \in (0, \infty) \), \( f(2x) = 2f(x) \);
2. For \( x \in (1, 2] \), \( f(x) = 2 - x \).

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

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