We know the definition of reverse pairs: for a permutation $$$a_1,a_2,\dots,a_n$$$ of length $$$n$$$, if there exists $$$i \lt j$$$ and $$$a_i \gt a_j$$$, then $$$a_i,a_j$$$ form a reverse pair. A permutation of length $$$n$$$ contains each of the numbers $$$1$$$ to $$$n$$$ exactly once.

Finding the number of reverse pairs should not be a difficult task for you. Now, you are given a permutation $$$a_1,a_2,\dots,a_n$$$ of length $$$n$$$. For each pair satisfying $$$i \lt j$$$ and $$$a_i \gt a_j$$$ of reverse pairs, color the cell in the $$$n \times n$$$ grid at row $$$i$$$ and column $$$a_j$$$. Please determine how many colored connected components are there in the final grid. Connected means that if two colored cells share an edge, then we consider these two cells to be connected.

You are now in a magical land, and you have five different types of sprites around you. The ability value of the $$$i$$$-th sprite is $$$i$$$.

At this moment, a sprite summoner with a sprite of ability value $$$6$$$ passes by. This sprite with an ability value of $$$6$$$ looks very powerful, so you quickly ask him how to obtain this kind of sprite and obtain a method called "Link Summon" from him.

Link Summon: Each time, choose some sprites with ability values from $$$1$$$ to $$$5$$$. For each sprite, you can choose to assign a weight of $$$1$$$ or a weight of $$$x$$$ ($$$x$$$ is its ability value), and then obtain one sprite with the sum of the weights of these sprites. However, due to magical restrictions, the sum of the weights cannot exceed $$$6$$$.

Now you want to know how many sprites with an ability value of $$$6$$$ you can summon by linking the sprites in your possession.

Given a cubic lattice of size $$$N \times M\times L$$$ such that each lattice point $$$(i,j,k)$$$ is either white or black. The coordinates of the lattice point are all positive integers. Let the cost of flipping the color of lattice point $$$(i,j,k)$$$ be $$$a_{i,j,k}$$$. Find the minimum cost to flip the colors of lattice points such that the bottom left corner $$$(1,1,1)$$$ is black, the top right corner $$$(N,M,L)$$$ is white, and any two different lattice points $$$(i_1, j_1, k_1)$$$ and $$$(i_2,j_2,k_2)$$$ satisfy at least one of the following conditions.

• $$$i_1 \gt i_2$$$
• $$$j_1 \gt j_2$$$
• $$$k_1 \gt k_2$$$
• $$$(i_1, j_1, k_1)$$$ is black.
• $$$(i_2, j_2, k_2)$$$ is white.

Given a grid with $$$n$$$ rows and $$$m$$$ columns, where rows are numbered from $$$1$$$ to $$$n$$$ from top to bottom, and columns are numbered from $$$1$$$ to $$$m$$$ from left to right. Now there are infinite $$$1+1$$$ L-shape tiles (as shown in the figure below), you need to use these tiles to cover the grid, so that only the cell in the $$$1$$$-st row and $$$m$$$-th column (located at the top right corner) is not covered, and each other cell is covered by exactly one tile. You need to determine if it is possible to achieve a covering that satisfies these conditions, and if so, output one covering scheme.

In this problem, you need to implement the output checker from the previous problem.

Given a grid with $$$n$$$ rows and $$$m$$$ columns, where rows are numbered from $$$1$$$ to $$$n$$$ from top to bottom, and columns are numbered from $$$1$$$ to $$$m$$$ from left to right. Now, there are infinite $$$1+1$$$ L-shaped tiles (as shown in the figure below), and you have used some tiles to cover the grid, but you do not know if you have covered the grid correctly.

Your covering scheme can be represented as a string of length $$$m$$$ for each of the $$$n$$$ rows. The string contains only the six characters UDLRC., where the $$$j$$$-th character of the $$$i$$$-th row represents the covering situation of the grid in the $$$i$$$-th row and the $$$j$$$-th column. The character . represents that the cell is not covered. If there is only one . in the covering scheme, and this . is located in the $$$1$$$-st row and the $$$m$$$-th column, then the covering scheme is correct; otherwise, it is incorrect. The character C represents the center of the tile (i.e., the bottom-left corner of the tile in the figure). The characters UDLR respectively represent that the top, bottom, left, right cell of this cell is covered by the center of the tile. If all L-shaped tiles are complete and each cell is covered by only one tile, then the covering scheme is correct; otherwise, it is incorrect.

Given some covering schemes, please determine if these schemes are correct.

Isoball is a challenging and fun game, and in this problem, we consider its 2D version. Given the radius of a circle and its initial position, there is a rectangle on the plane, and your goal is to let the circle be in the interior of the rectangle. To do this, you determine the direction of movement for the circle (represented by a vector). Now, you want to know if there exists a moment (including the initial moment) when the circle moves in this direction to reach the goal.

For a circle with center at $$$(p,q)$$$ and radius $$$r$$$, it is considered to be inside a rectangle with bottom-left corner at $$$(l_x,l_y)$$$ and top-right corner at $$$(r_x,r_y)$$$ if and only if $$$\forall p\in\{(x,y)\mid (x-p)^2+(y-q)^2\le r^2\}$$$, it holds that $$$l_x\le p_x\le r_x$$$ and $$$l_y\le p_y\le r_y$$$, where $$$p_x,p_y$$$ are the horizontal and vertical coordinates of point $$$p$$$, respectively.

Given $$$n$$$ integers $$$x_1,x_2,\ldots,x_n$$$. There are $$$q$$$ queries, each query gives a function $$$f(x)=(a\oplus x)-b$$$, please determine if there exists $$$1\le i \lt n$$$ such that $$$f(x_i)\cdot f(x_{i+1})\le 0$$$, if so, output a satisfying $$$i$$$, otherwise output $$$-1$$$.

$$$a\oplus b$$$ represents the bitwise XOR operation between $$$a$$$ and $$$b$$$.

GG and YY are playing a game on a pile of stones of size $$$n$$$. GG and YY play in turns. GG moves first. In each turn, the corresponding player can remove 1 or 2 stones from the pile. The player who cannot move loses. Both players want to win and get as many stones as possible. Assume GG and YY both follow the optimal strategy. Please decide the winner and answer the number of stones, $$$v$$$, that the winner gets at the end.

In a busy port, a large number of containers need to be stacked on the deck of cargo ships every day. For convenience, you can consider both the deck and the containers as rectangles parallel to the coordinate axes. The deck is a large rectangular area with the lower left corner at $$$(0,0)$$$ and the upper right corner at $$$(l,h)$$$.

You are a container scheduler and need to arrange the position of each container on the deck. In theory, you need to schedule it reasonably to maximize space utilization. However, you feel that you are underpaid, so you want to be lazy in the scheduling method. The scheduling method you provide is as follows:

• Due to height restrictions, containers can only be stacked in a single layer without overlapping.
• Since you don't have the budget to buy rotating cranes, containers cannot be rotated when placed.
• Containers follow the first-come, first-served order, meaning that containers at the front are placed first, and containers at the back are placed later, without changing the order of the containers. If a container cannot be placed during the process, it is discarded.
• When selecting a position to place a container, prioritize the position with the smallest $$$x$$$ coordinate that can accommodate it. If there are multiple candidate positions, choose the one with the smallest $$$y$$$ coordinate.

There are a total of $$$n$$$ containers, and for each container, the length along the $$$x$$$ axis and the width along the $$$y$$$ axis are given in order. Your task is to determine whether each container can be placed, and if so, provide the position of its lower left corner.

There are seven characters in Roman numerals: I, V, X, L, C, D, M, representing $$$1$$$, $$$5$$$, $$$10$$$, $$$50$$$, $$$100$$$, $$$500$$$, $$$1000$$$ respectively.

Now you will use Roman numerals and Arabic numerals to represent a positive integer in decimal. For example, $$$\texttt{X}2 = 10\times 10^1 + 2 \times 10^0=102$$$, $$$\texttt{IV}=1\times 10^1+5\times 10^0=15$$$.

For each digit, there is a certain cost for its representation. Find the minimum cost to represent a positive integer using a mixed representation.

You now have $$$m$$$ types of elements. The name of the first element is a, the second one is b, and so on. You will use these elements to attack monsters.

Specifically, you have a sequence of length $$$n$$$ for using elements, and you will use these elements to attack monsters in order. Assuming the current type of element is $$$x$$$, if the monster does not have an element attached to it, then the element $$$x$$$ will be attached to the monster. If the monster already has a certain element $$$y$$$, then the two elements will react, causing damage of $$$a_{y,x}$$$. After the reaction, only the element $$$x$$$ will remain on the monster, and reactions will also occur between elements of the same type.

However, now some types of elements may have become inert, meaning that these elements will not react or attach to the monster when used to attack, and will be ignored directly.

But you do not know which types of elements have become inert, so you want to know for each type of element, in the $$$2^m$$$ possible scenarios of whether it is inert or not, the total damage caused by the element sequence.

Students at the Trellis Hot Pot School are currently engaged in a heated discussion about which ingredients can be cooked in which type of hot pot.

There are two types of hot pots: spicy hot pots and soup pots. There are $$$n$$$ types of ingredients, numbered from $$$1$$$ to $$$n$$$. The cooking times for the $$$i$$$-th ingredient in the spicy hot pot and soup pot are denoted as $$$a_i$$$ and $$$b_i$$$, respectively (it is guaranteed that all $$$a_i$$$ and $$$b_i$$$ are distinct). Based on the discussion results, they have determined which ingredients can be cooked in each type of hot pot.

The students at the Trellis Hot Pot School have a unique way of eating hot pots. They will wait until all $$$n$$$ types of ingredients are ready and then decide whether each ingredient should be placed in the spicy hot pot or soup pot. If an ingredient can be placed in both pots, they will choose to place it in the pot with the shorter cooking time. After the discussion, all ingredients are put into the regarding pot. Due to the exhausting discussion process, when an ingredient is cooked, they will immediately fish it out and eat it.

The students at the Trellis Hot Pot School want to know the order in which the ingredients are fished out from the spicy hot pot and soup pot, respectively. Please output the answers in the order of fishing out time.