
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 the condition \(a_{k}=\frac{a_{1}+a_{2}+\ldots + a_{k-1}+ a_{k+1}+\ldots + a_{n}}{k+1} = \frac{120-a_{k}}{k+1}\) for all \(k=1,2,\ldots,n-1\), derive \(a_{k} = \frac{120}{k+2}\). Given that \(a_{k}\) are integers, determine the maximum value of \(n\). Calculate the values of \(a_{1}\), \(a_{2}\), \(a_{3}\), \(a_{4}\), and find the remaining value \(a_{5} = 120 - (a_{1} + a_{2} + a_{3} + a_{4})\).

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