
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  $ 19$  flags, of which  $ 10$  are identical blue flags, and  $ 9$  are identical green flags. Let  $ N$  be the number of distinguishable arrangements using all of the flags in which each flagpole has at least one flag 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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