
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 a chess board ( $8*8$ ) there are written the numbers  $1$  to  $64$ : on the first line, from left to right, there are the numbers  $1, 2, 3, ... , 8$ ; on the second line, from left to right, there are the numbers  $9, 10, 11, ... , 16$ ;etc. The  $\"+\"$  and  $\"-\"$  signs are put to each number such that, in each line and in each column, there are  $4$   $\"+\"$  signs and  $4$   $\"-\"$  signs. Then, the  $64$  numbers are added. Find all the possible values of this sum.

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