
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 a 6 x 4 grid (6 rows, 4 columns), 12 of the 24 squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column.  Let $N$ be the number of shadings with this property.  Find the remainder when $N$ is divided by 1000.
[AIME I 2007-10.png](https://artofproblemsolving.com/wiki/index.php/File:AIME_I_2007-10.png)

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