
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 ${a}_{n}=log_{2}(\frac{n+2}{n+1})$. A positive integer $k$ is defined as "good" if the sum of the first $k$ terms of the sequence $\{a_{n}\}$ is a positive integer. Find the sum of all "good" numbers within the range $\left[1,2023\right]$.

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