
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
On the board we write a series of  $n$  numbers, where  $n \geq 40$ , and each one of them is equal to either  $1$  or  $-1$ , such that the following conditions both hold:

(i) The sum of every  $40$  consecutive numbers is equal to  $0$ .
(ii) The sum of every  $42$  consecutive numbers is not equal to  $0$ .

We denote by  $S_n$  the sum of the  $n$  numbers of the board. Find the maximum possible value of  $S_n$  for all possible values of  $n$ .

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