
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  $k$  be a positive integer with the following property: For every subset  $A$  of  $\{1,2,\ldots, 25\}$  with  $|A|=k$ , we can find distinct elements  $x$  and  $y$  of  $A$  such that  $\tfrac23\leq\tfrac xy\leq\tfrac 32$ .  Find the smallest possible value of  $k$ .

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