
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
In a notebook, there are \( n \) integers written in descending order \( a_1 > a_2 > \ldots > a_n \) with a sum of 420. It is known that the \( k \)-th number \( a_k \), except the last one \( a_n \), is \( (k+1) \) times smaller than the sum of all the other written numbers. Find the maximum possible number \( n \) under these conditions and determine these numbers for the maximum possible \( n \).

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