
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
Given that \(\sum_{i=1}^{n} a_i \leqslant n\) and \(\sum_{j=1}^{n} b_j \leqslant n\), find the minimum value of
$$
\sqrt{2} \sum_{i=1}^{n} \sum_{j=1}^{n} \sqrt{\frac{1}{a_i}+\frac{1}{b_j}}-\sum_{i=1}^{n} \sum_{j=1}^{n} \sqrt{\frac{a_i^2+b_j^2}{a_i^2 b_j + a_i b_j^2}}.
$$

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