Problem K
King's Decree

SoCCat the Cat King is a very fair cat. Currently, SoCCat wants to ensure all cats can live in peace and harmony!

The Cat Kingdom is a very large kingdom, with $n$ cats living in it. The cats are numbered from $1$ to $n$.

King SoCCat has noticed a glaring problem in the kingdom — the uneven distribution of wealth among the cats. To solve this problem, King SoCCat has decreed that wealthy cats should give some of their wealth to poorer cats — to ensure that all cats can live in peace and harmony.

After surveying each cat in the kingdom, King SoCCat has determined that:

  • Cat $i$ has $w_ i$ units of wealth.

  • Cat $i$ is not willing to have less than $l_ i$ ($l_ i \leq w_ i$) units of wealth.

As a fair king, SoCCat wants to ensure that all cats have at least $l_ i$ units of wealth, and the poorest cat has the maximum possible wealth.

To achieve this, King SoCCat can do the following any number of times:

  • Choose a cat $i$ with wealth strictly greater than $l_ i$.

  • Choose any other cat $j$, and give $1$ unit of wealth from cat $i$ to cat $j$.

Unfortunately, King SoCCat is not very good at math. Therefore, as a friend of the Cat Kingdom, help King SoCCat!


The input starts with a line containing an integer $C$, denoting the number of test cases ($1 \leq C \leq 300\, 000$). Each test case has the following format.

The first line contains an integer $n$, denoting the number of cats in the kingdom ($1 \leq n \leq 300\, 000$). The sum of $n$ over all test cases does not exceed $300\, 000$.

The second line contains $n$ integers $w_1, w_2, \ldots , w_ n$, denoting the wealth of each cat ($1 \leq w_ i \leq 10^9$).

The third line contains $n$ integers $l_1, l_2, \ldots , l_ n$, denoting the minimum wealth that each cat is willing to have ($1 \leq l_ i \leq w_ i$).


For each test case, output a line containing a single integer denoting the maximum possible wealth that the poorest cat can have, if King SoCCat distributes the wealth of the cats in the kingdom fairly and optimally.

Sample Input 1 Sample Output 1
4 2 3 6 12
3 2 3 3 10
998 244 353 3109 4650 4305 6507 9699
100 200 300 2040 4230 4236 5234 3233

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