
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 \(a\), \(b\), and \(n\) be positive integers. A lemonade stand owns \(n\) cups, all of which are initially empty. The lemonade stand has a *filling machine* and an *emptying machine*, which operate according to the following rules:      
- If at any moment,         \(a\) completely empty cups are available,         the filling machine spends the next \(a\) minutes         filling those \(a\) cups simultaneously and doing nothing else.
- If at any moment, \(b\) completely full cups are available,         the emptying machine spends the next \(b\) minutes         emptying those \(b\) cups simultaneously and doing nothing else.
 Suppose that after a sufficiently long time has passed, both the filling machine and emptying machine work without pausing. Find, in terms of \(a\) and \(b\), the least possible value of \(n\).

*Proposed by Raymond Feng*

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