
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
The 30th Olympic Games were held in London. Let the sequence $a_n = \log_{n+1}(n+2)$ ($n \in \mathbb{N}^*$), and define a real number $k$ as an Olympic lucky number if the product $a_1 \cdot a_2 \cdot a_3 \cdots a_k$ is an integer. Then, the sum of all Olympic lucky numbers in the interval $[1, 2012]$ is ______.

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