
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 Zumingle, all words consist only of the letters $N, E,$ and $T$. $E$ is a vowel and $N$ and $T$ are consonants. A string of $N's, E's,$ and $T's$ is a word in Zumingle if and only if between any two $E's$ there appear at least three consonants. Let $M$ denote the number of $12$-letter Zumingle words. Determine the remainder obtained when $M$ is divided by $1000$.

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