
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
Find all integers \(a, b, c\) such that for any real numbers \(u, v\) with \(0 \leq u < v \leq 1\), there exists a positive integer \(n\) making \(a n^2 + b n + c \geq 0\),

and \(\{\sqrt{a n^2 + b n + c}\} \in (u, v)\).

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