
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
Let \( a_{1}, a_{2}, \cdots, a_{k} \) be \( k \) different positive integers, none of which contain any square factors (one 1 is allowed). Then, for any non-zero rational numbers \( b_{1}, b_{2}, \cdots, b_{k} \), we have \( \sum_{i=1}^{k} b_{i} \sqrt{a_{i}} \neq 0 \). Note: A positive integer \( n \) is said to not contain square factors if there does not exist an integer \( d > 1 \) such that \( d^2 \) divides \( n \).

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