
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
Let  $n$  and  $k$  be positive integers such that  $k\leq 2^n$ . Banana and Corona are playing the following variant of the guessing game. First, Banana secretly picks an integer  $x$  such that  $1\leq x\leq n$ . Corona will attempt to determine  $x$  by asking some questions, which are described as follows. In each turn, Corona chooses  $k$  distinct subsets of  $\{1, 2, \ldots, n\}$  and, for each chosen set  $S$ , asks the question  "Is  $x$  in the set  $S$ ?''.

Banana picks one of these  $k$  questions and tells both the question and its answer to Corona, who can then start another turn.

Find all pairs  $(n,k)$  such that, regardless of Banana's actions, Corona could determine  $x$  in finitely many turns with absolute certainty.

*Pitchayut Saengrungkongka, Thailand*

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