
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
Let \( N \) be the set of natural numbers, \( R \) be the set of real numbers, and \( S \) be the set of functions \( f: N \rightarrow R \) 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 \( n = 1, 2, \ldots \)

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

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