
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 $a_n=\log_{n+1}(n+2)$, $n∈\mathbb{N}^*$, find the sum of all deficient numbers $k$ in the interval $(1,62)$, where a deficient number is defined as a value of $k$ that makes the product $a_1⋅a_2⋅a_3⋅a_4…a_k$ an integer.

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