
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
A natural number  $k > 1$  is called *good* if there exist natural numbers  $$ a_1 < a_2 < \cdots < a_k $$  such that  $$ \dfrac{1}{\sqrt{a_1}} + \dfrac{1}{\sqrt{a_2}} + \cdots +  \dfrac{1}{\sqrt{a_k}} = 1 $$  .

Let  $f(n)$  be the sum of the first  $n$  *[good* numbers,  $n \geq$  1. Find the sum of all values of  $n$  for which  $f(n+5)/f(n)$  is an integer.

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