
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
Numbers  $0, 1$  and  $2$  are placed in a table  $2005 \times 2006$  so that total sums of the numbers in each row and in each column are factors of  $3$ . Find the maximal possible number of  $1$ 's that can be placed in the table.

*(6 points)*

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