
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
Given the sequence $\{a_n\}$ where $a_n = \log_{(n+1)}{(n+2)}$ (here $n$ belongs to the set of positive integers $N^*$), we define an integer $n$ as a "preferred number" if the product $a_1 \cdot a_2 \cdot a_3 \cdots a_n$ is an integer. The sum of all "preferred numbers" in the interval $(1, 2012]$ is ______.

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