
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
Let $a_n = \log_{(n+1)}(n+2)$ where $n \in \mathbb{N}$. We define a number $n$ as an "inferior number" if the product $a_1a_2…a_n$ is an integer. Find the sum of all inferior numbers in the interval $(1, 2004)$.

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