
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

Two people are playing a game. One person thinks of a natural number \( n \), and the other asks questions such as "Is it true that \( n \) is not less than \( x \)?" (where the number \( x \) is chosen at their discretion) and receives the answers "yes" or "no." For any possible strategy \( T \) of the second player, we associate a function \( f_{T}(n) \), which is the number of questions asked (until the number is guessed) if the number \( n \) was the one thought of. Suppose, for instance, that strategy \( T \) consists of first asking the questions: "Is it true that \( n \) is not less than 10?", "Is it true that \( n \) is not less than 20?", ... until the answer "no" is received to some question "Is it true that \( n \) is not less than \( 10(k+1) \)?". Then, they ask questions like "Is it true that \( n \) is not less than \( 10k+1 \)?", "Is it true that \( n \) is not less than \( 10k+2 \)?" and so on. In this case, \( f_{T}(n)=a+2+(n-a)/10 \), where \( a \) is the last digit of the number \( n \), meaning \( f_{T}(n) \) grows approximately like \( n/10 \).

a) Propose a strategy for which the function \( f_{T} \) grows more slowly.

b) When comparing two strategies, it is convenient for any arbitrary strategy \( T \) to introduce a function \( \bar{f}_{T} \), whose value for any natural number \( n \) is the maximum of the numbers \( f_{T}(k) \), where \( k \) takes values from 1 to \( n \). Provide a lower bound for \( \bar{f}_{T} \) for any given strategy \( T \).

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