
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
Given $m, n, k \in N$, there are 4 glasses with capacities $m$, $n$, $k$, and $m+n+k$ liters respectively. The following operations are allowed: pouring all the wine from one glass into another or pouring wine from one glass into another until it is full. Initially, the large glass is full of wine and the other 3 glasses are empty. What is the necessary and sufficient condition on $m, n, k$ so that for any $s \in N, s < m + n + k$, it is possible to have exactly $s$ liters of wine in one of the glasses after a series of operations?

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