
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
There are two distinguishable flagpoles, and there are $23$ flags, of which $12$ are identical blue flags, and $11$ are identical green flags. Let $N$ be the number of distinguishable arrangements using all of the flags in which each flagpole has at least 3 flags and no two green flags on either pole are adjacent. Find the remainder when $N$ is divided by $1000$.

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