
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
A 0-1 sequence of length  $2^k$  is given. Alice can pick a member from the sequence, and reveal it (its place and its value) to Bob. Find the largest number  $s$  for which Bob can always pick  $s$  members of the sequence, and guess all their values correctly.

Alice and Bob can discuss a strategy before the game with the aim of maximizing the number of correct guesses of Bob. The only information Bob has is the length of the sequence and the member of the sequence picked by Alice.

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