
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  $n,k$  be positive integers such that  $n\geq2k>3$  and  $A= \{1,2,...,n\}.$  Find all  $n$  and  $k$  such that the number of  $k$ -element subsets of  $A$  is  $2n-k$  times bigger than the number of  $2$ -element subsets of  $A.$ 

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