
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 the largest constant \( C \) such that for any positive integer \( n \), if \( \left\{a_{i}, b_{i}\right\}(1 \leqslant i \leqslant n) \) are \( n \) distinct pairs of positive integers, then
$$
\sum_{i=1}^{n} \max \left\{a_{i}, b_{i}\right\} \geqslant C n \sqrt{n},
$$
where \( \max \{x, y\} \) denotes the maximum of \( x \) and \( y \).

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