
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 sequence $\{a_{n}\}$ where $a_{n}=\left\{\begin{array}{l}{1, n=1}\\{\log_{n}(n+1), n\geq 2, n\in \mathbb{N}^{*}}\end{array}\right.$, a positive integer $k$ is defined as a "happy number" if $a_{1}\cdot a_{2}\cdot a_{3}\cdot \ldots \cdot a_{k}$ is an integer. Find the sum of all "happy numbers" in the interval $\left[1, 2022\right]$.

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