
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
An integer is square-free if it is not divisible by  $a^2$  for any integer  $a>1$ . Let  $S$  be the set of positive square-free integers. Determine, with justification, the value of\[\sum_{k\epsilon S}\left[\sqrt{\frac{10^{10}}{k}}\right]\]where  $[x]$  denote the greatest integer less than or equal to  $x$ 

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