Kaguya gives you two arrays $$$[a_1,a_2...a_n]$$$ and $$$[b_1,b_2...b_m]$$$, and you need to select two subsequences $$$[a_{c_1},a_{c_2}...a_{c_p}]$$$ and $$$[b_{d_1},b_{d_2}...b_{d_q}]$$$ that satisfy the following conditions:

• $$$c_1\le 2,\ d_1\le 2$$$;
• $$$\forall i \ge 2, c_i-c_{i-1}\le 2,\ d_i-d_{i-1}\le 2$$$;
• $$$c_p=n,\ d_q=m$$$.

Now she asks you whether no matter how you choose $$$[c_1,c_2...c_p]$$$, there exists a $$$[d_1,d_2...d_q]$$$ such that $$$[b_{d_1},b_{d_2}...b_{d_q}]$$$ is a subsequence of $$$[a_{c_1},a_{c_2}...a_{c_p}]$$$.

123456zmy is playing GT: New Horizons and wants to build $$$n$$$ Bricked Blast Furnaces. A Bricked Blast Furnace is a multiblock structure. In this problem, we only consider the layer containing the Bricked Blast Furnace block.

Bricked Blast Furnace and the layer containing the block.

We use 'B', 'F', and 'L' (or their lowercase) to represent Bricked Blast Furnace, Firebricks, and Lava, respectively. A complete Bricked Blast Furnace structure consists of 7 Firebricks, 1 Lava, and 1 Bricked Blast Furnace. The structure can be rotated arbitrarily, as shown below.

Multiple structures can share the same Firebricks 'F'. However, sharing 'B' or 'L' is not allowed. For example, there are 4 complete Bricked Blast Furnace structures below.

123456zmy wants to use no more than $$$3\times10^5$$$ Firebricks to form at least $$$n$$$ Bricked Blast Furnace structures. Please tell him how to arrange the blocks.

$$$~\\$$$

Deep in the Arctic permafrost lies an obsidian monolith etched with glowing celestial runes. These symbols — said to hold the fabric of spacetime — shift positions during auroral storms. Ancient texts warn that if the runes ever freeze in a "disordered sequence," two must be swapped exactly once to restore cosmic balance. Any additional tampering would collapse reality into quantum limbo.

When Vistar discovers the monolith frozen as a sequence, its hum resonates with impending doom. The decoded notes reveal: "One swap averts the rupture. Two swaps lead to chaos."

Now, as the ground trembles, he must answer: Are the runes already ordered by fate's design? If not, do exactly two exist whose swap would align them?

More formally, given $$$n$$$ numbers, please determine:

• Are the numbers already in ascending order?
• If not, can swapping two of them restore ascending order?

Keine is a teacher in Touhou Eiyashou. One day she gives you a problem in her class:

Give you an array $$$a_i$$$, we define $$$s_i$$$ as the prefix sum of $$$a_i$$$ (In other words, $$$s_i=\sum^i_{j=1}a_j$$$). Assume that an array $$$a_i$$$ is $$$\textbf{good}$$$ if and only if each $$$s_i$$$ satisfies $$$s_i \ge 0$$$.

Now Keine will perform a cyclic placement operation for $$$n$$$ times: Place $$$a_1$$$ behind $$$a_n$$$, which means that $$$[a_1,a_2...a_{n-1},a_n]$$$ will change to $$$[a_2,a_3...a_n,a_1]$$$. Then Keine will ask you whether the array $$$a_i$$$ is $$$\textbf{good}$$$.

Please output the number of $$$\textbf{good}$$$ arrays.

$$$~\\$$$

As we all know, Neuro-sama is the best OSU! player.

After defeating every challenger, Neuro-sama wants to play another game named USO!. The rules of USO! are as follows:

There is an array $$$a$$$ of length $$$n$$$, with indices starting from $$$0$$$. You need to fill the array with a permutation of integers from $$$0$$$ to $$$n-1$$$ (i.e., each number is used only once).

You are given a binary string $$$s$$$ of length $$$n$$$. The array $$$a$$$ must satisfy the following conditions for all indices $$$i\ (0 \le i \lt n)$$$:

• If $$$s_i = 0$$$, then $$$a_i$$$ must be less than $$$a_{(i+1)\%n}$$$;
• If $$$s_i = 1$$$, then $$$a_i$$$ must be greater than $$$a_{(i+1)\%n}$$$.

Neuro-sama wants to play USO! with you. Please output any legal $$$a$$$ as soon as possible. If there is no legal combination, output $$$-1$$$.

$$$~\\$$$

Vistar finds himself trapped in a mysterious maze consisting of $$$n$$$ rooms. Surprisingly, the maze has no traditional doors — Instead, the $$$i-$$$th room contains $$$a_i$$$ portals.

Each portal is uniquely paired with exactly one other portal in the maze, forming a bidirectional connection. When Vistar enters a portal, he is instantly transported to the room of its paired portal. In particular, the portal may be paired to another portal within the same room.

Vistar begins his journey in room $$$1$$$ and aims to reach room $$$n$$$. However, the specific pairings of the portals are unknown. Your task is to determine whether Vistar can always reach the exit, regardless of how the portals are paired.

$$$~\\$$$

Once again, Vistar finds himself trapped in a mysterious maze consisting of $$$n$$$ rooms. Surprisingly, the maze has no traditional doors — Instead, the $$$i-$$$th room contains $$$a_i$$$ portals.

Each portal is uniquely paired with exactly one other portal in the maze, forming a bidirectional connection. When Vistar enters a portal, he is instantly transported to the room of its paired portal. In particular, the portal cannot be paired to another portal within the same room.

Vistar begins his journey in room $$$1$$$ and aims to reach room $$$n$$$. However, the specific pairings of the portals are unknown. Your task is to determine whether Vistar can always reach the exit, regardless of how the portals are paired.

Hakurei Reimu is good at playing maimai DX.

In maimai DX, there are $$$8$$$ touch-sensitive buttons arranged in a circular layout around the screen. We number them from $$$1$$$ to $$$8$$$ (picture 1).

picture 1

One day, Reimu is studying simai language. We always use simai language to describe a slide. A slide contains three parts: starting point, ending point, and sliding path.

There are 4 kinds of slides in maimai DX:

• $$$x-y$$$: The segment from $$$x$$$ to $$$y$$$ (picture 2 indicates $$$1-4$$$);
• $$$x \gt y$$$: The circular arc clockwise from $$$x$$$ to $$$y$$$ (picture 3 indicates $$$2 \gt 5$$$);
• $$$x \lt y$$$: The circular arc anticlockwise from $$$x$$$ to $$$y$$$ (picture 4 indicates $$$3 \lt 6$$$);
• $$$x\lor y$$$: First the segment from $$$x$$$ to the center point, then the segment from the center point to $$$y$$$ (picture 5 indicates $$$1\lor 3$$$).

For example, there are $$$4$$$ slides in picture 6: $$$1 \gt 3,3\lor 5,1 \lt 7,7\lor 5$$$.

picture 6

Now we assume the radius of the maimai DX circle is $$$50$$$cm. Given a slide $$$x?y$$$, can you tell me the length of the slide?

$$$~\\$$$

Three times Vistar dreamed of the marvellous jewel, and three times was he snatched away while still he paused on the high terrace above it.

Each vision just ending the same way... In dreaming, Vistar had spent months planning the ultimate theft — to steal the Legendary Jewel of Mathematical Truth, a gem said to grant its wielder infinite knowledge.

The jewel was kept atop the Crown of Proofs, guarded by the relentless Math Police, an elite order devoted to protecting the foundations of mathematics. Their stronghold is The Tree of Mathematics, a rooted tree with $$$n$$$ nodes, where every branch represented a different field — number theory, graph theory, topology, and beyond.

On the grand night, Vistar infiltrated the root (node $$$1$$$). But as he reached for the jewel, an alarm echoed through the tree. Immediately, he found himself surrounded by the guards — At every leaf of this tree stood a Math Police officer, armed with Group Theory Rifles and Algebra Theory Snipers, determined to stop him from stealing the jewel atop the Crown of Proofs.

The scene was so vivid and clear:

• Each turn, Vistar could move one step to an adjacent node (or choose to stay);

• Then, all officers would simultaneously move one step along their respective shortest paths toward the root. If an officer was already on the root, then he would stay still;

• If at any moment (for example, when Vistar has finished moving but the officers have not) Vistar shared a node with any officer, the chase ends — and so does his freedom.

Waking in a cold sweat, Vistar couldn't shake the question: Under such a siege, was escape even possible? We define escape as possible if Vistar isn't captured after $$$n$$$ turns.

Since he is in a panic, he wants you to answer the question.

In the study of probability theory and stochastic processes, the Random Descent Process (RDP) is an important class of mathematical models. In this context, we examine the following problem:

Given a positive integer $$$n$$$. You start with an integer $$$m$$$, whose initial value is $$$n$$$. In each operation, you replace $$$m$$$ with a uniformly random integer between $$$0$$$ and $$$m-1$$$ (inclusive) until $$$m$$$ becomes $$$0$$$. What is the expected number of operations required?

$$$~\\$$$

Be there, or be squared!

In the vast, shimmering expanse of the digital stream, where data flows like rivers of light, Neuro-sama and her mischievous sister, Evil, are locked in a high-stakes game of hide-and-seek. The stream is a labyrinth of $$$n$$$ nodes, each glowing with a unique identifier from $$$1$$$ to $$$n$$$, pulsing with electric pathways.

For any node $$$u$$$, the digital pathways twist and turn under the following rules:

• If $$$u$$$ is a perfect square, it has a directed edge towards $$$\sqrt{u}$$$;
• Otherwise, if $$$u \le n - 3$$$, it has a directed edge towards $$$u + 3$$$.

Evil has hidden herself, leaving only $$$q$$$ suspicious locations. Now, Neuro wonders: for each possible hiding spot $$$x$$$, how many nodes in the network (from $$$1$$$ to $$$n$$$) can reach it?

Since the number may be large, she wants to output it modulo $$$998244353$$$.

@atari_desu

The casual game of hide-and-seek from the past has now escalated into a storm within the digital stream.

In their last game, Neuro-sama, with her exceptional computational abilities, could always swiftly pinpoint her sister Evil's hiding spots. However, being found so easily time and again ignited a flame of defiance in Evil's heart.

The once shimmering labyrinth of $$$n$$$ nodes has, under Evil's influence, expanded and contorted dramatically, becoming a vast and far more intricate "data cosmos." The rules of connection between nodes remain thesame, like the immutable physical laws of this universe.

For any node $$$u$$$:

• If $$$u$$$ is a perfect square, it has a directed edge towards $$$\sqrt{u}$$$;
• Otherwise, if $$$u \le n - 3$$$, it has a directed edge towards $$$u + 3$$$.

Facing a digital cosmos of unprecedented scale and peril, Neuro-sama understands that tracing every possible path for each suspicious location is no longer feasible. She needs to rapidly calculate: for each possible hiding spot $$$x$$$, how many nodes in the network (from $$$1$$$ to $$$n$$$) can reach it.

Since the number may be large, she wants to output it modulo $$$998244353$$$.