
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 the set of natural numbers \(\mathbf{N}\), the set of real numbers \(\mathbf{R}\), and the set \(S\) of functions \(f: \mathbf{N} \rightarrow \mathbf{R}\) that satisfy the following two conditions:
1. \(f(1) = 2\)
2. \(f(n+1) \geq f(n) \geq \frac{n}{n+1} f(2n)\) for \(n = 1, 2, \cdots\)

Find the smallest natural number \(M\) such that for any \(f \in S\) and any \(n \in \mathbf{N}\), \(f(n) < M\).

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