One day on a rainy morning, Tahura was enjoying a cup of tea on her balcony when she noticed that some bricks were laying on the waterway trapping the rainwater that the waterway was supposed to drain. All the bricks were square-shaped and some were stacked on top of another. Now she wanted to see how much water it can trap after it had rained for several hours.

She made a few assumptions —

• The drain has infinite depth.
• The width of each brick is $$$1$$$.
• The water beyond the ends on both sides of the brick structure is discarded. More specifically she only measured the water trapped between the first and the last brick.

Then she represented the brick structure with an array of integers. Each element of the array represents a stack of bricks and equals the number of bricks in that stack.

If The elevation map of the brick structure looked like this, the corresponding array would look like $$$[0,1,0,2,1,0,1,3,2,1,2,1]$$$.

Given the array representing the elevation map, can you find the maximum amount of water that could be trapped in the brick structure for her?

There are $$$n$$$ offices and you need to get some signatures from each of them to complete a single form. You're given an array $$$a_1, a_2, \dots, a_n$$$ representing the number of signatures needed from the $$$i$$$-th place to complete the form. You're also given an array $$$b_1, b_2, \dots, b_n$$$ representing the maximum number of signatures in total you can get from the $$$i$$$-th place.

You can also go to the Boss and get at most $$$k$$$ special signatures from him. Each of these signatures has the power to act as a substitute for any other required signature. $$$1$$$ special signature can be used in place of exactly $$$1$$$ other required signature.

Find out the maximum number of forms you can complete.

You're given an array $$$a_1, a_2, \dots, a_n$$$. You need to pair the elements of the array with each other into $$$\frac{n}{2}$$$ pairs. The length of the array $$$n$$$ is an even number.

The cost of a pair $$$(a_i, a_j)$$$ is $$$\max(a_i, a_j)$$$. You need to find the minimum possible sum of costs of the $$$\frac{n}{2}$$$ pairs, over all the possible ways to pair the elements with each other.

You're given an array of integers $$$a_1, a_2, \dots, a_n$$$. You need to process $$$q$$$ queries. In each query, you're given an integer $$$b$$$. you need to find the number of divisors of $$$b$$$ that aren't present in the array $$$a$$$.

You're given an array of integers $$$a_1, a_2, \dots, a_n$$$ and an integer $$$m$$$. Find the number of divisors of $$$m$$$ that don't have any multiples in $$$a$$$.

In other words, count the number of integers $$$d$$$ satisfying -

• $$$d$$$ divides $$$m$$$.
• $$$d$$$ doesn't divide $$$a_i$$$ for $$$1 \leq i \leq n$$$.

You need to construct an array $$$a_1, a_2, \dots, a_n$$$ $$$\left(0 \leq a_i \lt 2^{30}\right)$$$ that satisfies $$$m$$$ given constraints. Each constraint is one of two types —

1. $$$a_i \neq x$$$, for given $$$i$$$ and $$$x$$$.
2. $$$a_i \oplus a_j = x$$$, for given $$$i,\ j$$$ and $$$x$$$.

Roshid is a university student who works hard to maintain his grades and excel in his classes. He spends long hours studying and taking notes, both in the library and at home on his laptop. Currently, Roshid is in the thick of midterm exam season. He has been working hard to prepare for his impending examinations for weeks, reading through textbooks and taking copious notes. Roshid has been drafting up study materials and practice problems on his laptop for countless hours, but he's now run into trouble. His keyboard's fifth row, which includes the keys for the letters z to m, has abruptly stopped functioning.

Roshid panics for a moment, thinking about all the time he's spent preparing for his exams. How will he be able to complete his assignment if he can't type the characters in the fifth row? How will he conquer this exam war? Roshid realizes that he'll need to fix the issue before he can proceed with his studies. However, before he can do that, he wonders what his work will look like if he types the rest of the writing without the keys functioning properly.

Saimon is a British Billionaire and he wanted to spend his last Christmas vacation in Northern Canada. So, he took a couple of his friends and sailed his yacht on a sunny day in December. But after four days, while passing through the North Atlantic Ocean his ship hit an iceberg and everyone but him died. Somehow, he managed to survive and reached a new island country. The island was so beautiful and full of magical events.

Since Saimon is a successful businessman, among all the magical events, a business model attracted him the most. While observing the model for a couple of weeks, he found some interesting facts about this model. A special type of currency named Emm coin is used for all the transactions in this small island country. By investing in a pair of Emm coins, from the fourth day and on, it is possible to make a profit of another pair of Emm coins. Saimon somehow managed $$$k$$$ pairs of Emm coins. Based on the information about the magical business model, how many Emm coins would he have after $$$n$$$ days?

He tried to find the answer, but failed. In fact, he is great in business but poor in Math. So, he contacted your team and asked you to solve this problem for him.

You have $$$n$$$ items in a vertical stack, the color of the $$$i$$$-th item from the top is $$$c_i$$$.

You need to process $$$q$$$ queries on this stack. In each query, you'll be given a color $$$d$$$ and you need to —

• Find the top-most item of color $$$d$$$ in the current stack and print its position from the top of the stack.
• And then, move this item to the top of the stack.

Masud found a map of the earth and is planning on visiting it. There are a total of $$$n$$$ cities. Each city belongs to only one country. Two cities are connected via one-way roads. Two cities are considered to be in the same country if it is possible to reach the second city (directly or indirectly) starting from the first one and also reach the first city starting from the second one. Otherwise, the cities are of different countries.

Now Masud is very busy and can only prepare the necessary documents to travel to at most two different countries. But he wants to visit as many cities as possible. Can you help him find the maximum number of cities he can visit by traveling to at most two countries?

Note that the two chosen countries do not necessarily need to be reachable from each other.

You're given a list of abilities as an array of strings $$$s_1, s_2, \dots, s_n$$$. All characters of each ability will be unique to each other. You want to build a spell using these abilities. To make a spell these abilities can be inserted, appended, or prepended to the spell in multiple iterations. For example, let's consider the abilities are abc, def, and pqrt.

1. "" (empty string)
2. "abc" (abc is appended/prepended in the empty string)
3. "adefbc" (def is inserted at index $$$1$$$ of the previous string)
4. "adefbcpqrt" (pqrt is appended in the previous string)
5. "pqrtadefbcpqrt" (pqrt is prepended in the previous string)
6. "pqrtadefabcbcpqrt" (abc is inserted at index $$$7$$$ of the previous string)
7. $$$\cdots$$$

And so on. The spells can be generated in many iterations. Each spell is valid except for the empty string at iteration $$$1$$$.

Now, given a spell, can you verify whether the spell is valid or not?