
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 $a_{n}=\log_{(n+1)}(n+2)$ $(n\in\mathbb{N}^*)$, define the product $a_{1}\cdot a_{2}\ldots a_{k}$ as an integer for $k(k\in\mathbb{N}^*)$ called "ideal number", then the sum of all ideal numbers in the interval $[1,2015]$ is ______.

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