
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 ${\{a_n\}}$ that satisfies $a_n = \log_{n+1}(n+2) \ (n \in \mathbb{N}^*)$, define $k \ (k \in \mathbb{N}^*)$ that makes the product ${a_1} \cdot {a_2} \cdot {a_3} \cdot \ldots \cdot {a_k}$ a positive integer as a "desired number". Find the sum of all "desired numbers" within the interval $[1, 2018]$.

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