
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
Find all integers  $k\ge 3$  with the following property: There exist integers  $m,n$  such that  $1<m<k$ ,  $1<n<k$ ,  $\gcd (m,k)=\gcd (n,k) =1$ ,  $m+n>k$  and  $k\mid (m-1)(n-1)$ .

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