
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
A notebook contains \( n \) integers, ordered in descending order \( a_{1} > a_{2} > \ldots > a_{n} \) and having a sum of 840. It is known that the \( k \)-th number \( a_{k} \), except the last one, i.e., for \( k \neq n \), is \( (k+1) \) times smaller than the sum of all other numbers. Find the maximum possible value of \( n \) under these conditions. Find these numbers for the maximum possible \( n \).

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