
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
In each cell of a  $4 \times 4$  table a digit  $1$  or  $2$  is written. Suppose that the sum of the digits in each of the four  $3 \times 3$  sub-tables is divisible by  $4$ , but the sum of the digits in the entire table is not divisible by  $4$ . Find the greatest and the smallest possible value of the sum of the  $16$  digits.

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