
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
1. Let \( \mathbb{N} = \{ 1, 2, 3, \ldots \} \) be the set of all natural numbers and \( f: \mathbb{N} \rightarrow \mathbb{N} \) be a function. Suppose \( f(1) = 1 \), \( f(2n) = f(n) \) and \( f(2n + 1) = f(2n) + 1 \) for all natural numbers \( n \).

(i) Calculate the maximum value \( M \) of \( f(n) \) for \( n \in \mathbb{N} \) with \( 1 \leq n \leq 1994 \).

(ii) Find all \( n \in \mathbb{N} \), with \( 1 \leq n \leq 1994 \), such that \( f(n) = M \).

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