
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 an arithmetic sequence \\(\{a_n\}\) where \\(a_1=31\\), and \\(S_n\\) is the sum of the first \\(n\\) terms, with \\(S_{10}=S_{22}\\).  
\\((1)\\) Find \\(S_n\\);  
\\((2)\\) How many terms of this sequence have the maximum sum, and what is this maximum value?

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