
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 two distinct flagpoles with $24$ flags total, consisting of $14$ identical blue flags and $10$ identical red flags. Let $M$ be the number of possible ways to arrange these flags such that each flagpole has at least one flag, no two red flags on either pole are adjacent, and each sequence starts with a blue flag. Calculate the remainder when $M$ is divided by 1000.

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