
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 play a game. One player thinks of a natural number \( n \), and the other asks questions of the type "is it true that \( n \) is not less than \( x \)" (the number \( x \) can be chosen at their discretion) and receives answers "yes" or "no." To each possible strategy \( T \) of the second player, we associate a function \( f_T(n) \), which equals the number of questions asked (before guessing) if the number \( n \) was thought of. For example, if the strategy \( T \) consists of first asking questions like "is it true that \( n \) is not less than 10?", "is it true that \( n \) is not less than 20?", etc., until to some question "is it true that \( n \) is not less than \( 10(k+1) \)?" the answer "no" is given, and then asking 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. Then \( f_T(n) = a+2+(n-a)/10 \), where \( a \) is the last digit of the number \( n \), so \( f_T(n) \) grows approximately as \( n/10 \).

a) Propose a strategy for which the function \( f_T \) grows more slowly.

b) When comparing two strategies, it is convenient for an arbitrary strategy \( T \) to introduce the function \( \bar{f}_T \) instead of the function \( f_T \), the value of which for any natural number \( n \) equals the maximum of the numbers \( f_T(k) \), where \( k \) ranges from 1 to \( n \). Estimate the lower bound of \( \bar{f}_T \) for an arbitrary strategy \( T \).

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