
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 a board of  $3 \times 3$  you want to write the numbers  $1, 2, 3, 4, 5, 6, 7, 8$  and a number in their boxes positive integer  $M$ , not necessarily different from the above. The goal is that the sum of the three numbers in each row be the same $a)$  Find all the values of  $M$  for which this is possible. $b)$  For which of the values of  $M$  found in  $a)$  is it possible to arrange the numbers so that no only the three rows add the same but also the three columns add the same?

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