
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

Consider the function \( f: \mathbb{N}_{0} \rightarrow \mathbb{N}_{0} \), where \( \mathbb{N}_{0} \) is the set of all non-negative integers, defined by the following conditions:
1. \( f(0) = 0 \),
2. \( f(2n) = 2f(n) \), and
3. \( f(2n+1) = n + 2f(n) \) for all \( n \geq 0 \).

(a) Determine the three sets \( L := \{ n \mid f(n) < f(n+1) \} \), \( E := \{ n \mid f(n) = f(n+1) \} \), and \( G := \{ n \mid f(n) > f(n+1) \} \).

(b) For each \( k \geq 0 \), find a formula for \( a_{k} := \max \left\{ f(n) : 0 \leq n \leq 2^{k} \right\} \) in terms of \( k \).

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