
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
\[ \sum_{k \in S}\left\lfloor\sqrt{\frac{10^{10}}{k}}\right\rfloor \text{,} \]
where \(\lfloor x\rfloor\) denotes the greatest integer less than or equal to \( x \). For example, \(\lfloor 2.5\rfloor = 2 \).

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