
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
Consider a  $2\times 3$  grid where each entry is either  $0$ ,  $1$ , or  $2$ . For how many such grids is the sum of the numbers in every row and in every column a multiple of  $3$ ? One valid grid is shown below:  $$ \begin{bmatrix} 1 & 2 & 0  2 & 1 & 0 \end{bmatrix} $$  

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