
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) Let  $ x_1, x_2, ..., x_n $  ( $ n> 2 $ ) be negative real numbers and  $ x_1 + x_2 + ... + x_n = m. $  
Determine the maximum value of the sum $ S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n. $ b) Let  $ x_1, x_2, ..., x_n $  ( $ n> 2 $ ) be nonnegative natural numbers and  $ x_1 + x_2 + ... + x_n = m. $  
Determine the maximum value of the sum $ S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n. $ 

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