The world is full of colors.

When creating an artwork, painters often use the technique of complementary colors to make their works more visually striking. In the $$$\mathtt{RGB}$$$ color model, complementary colors have a clear special property when represented as color codes.

Specifically, in the $$$\mathtt{RGB}$$$ color model, a color code consists of a $$$\mathtt{\#}$$$ followed by the hexadecimal representations of its $$$\mathtt{R}$$$ (red), $$$\mathtt{G}$$$ (green), and $$$\mathtt{B}$$$ (blue) values concatenated together. The values of $$$\mathtt{R}$$$, $$$\mathtt{G}$$$, and $$$\mathtt{B}$$$ all range from $$$[0, 255]$$$. For example, the color code $$$\mathtt{\#1f1e33}$$$ represents a very dark purple. Its $$$\mathtt{R}$$$, $$$\mathtt{G}$$$, and $$$\mathtt{B}$$$ values in hexadecimal are $$$1f$$$, $$$1e$$$, $$$33$$$, and in decimal are $$$31$$$, $$$30$$$, $$$51$$$. In particular, if any of the hexadecimal representations of $$$\mathtt{R}$$$, $$$\mathtt{G}$$$, or $$$\mathtt{B}$$$ has fewer than two digits, a leading $$$0$$$ must be added. For example, the color code for black is $$$\mathtt{\#000000}$$$.

We define two colors to be complementary colors if and only if their $$$\mathtt{R}$$$, $$$\mathtt{G}$$$, and $$$\mathtt{B}$$$ values satisfy that each corresponding pair sums to $$$255$$$.

Whalica landed on a colorless planet during her interstellar voyage. She wanted to bring more vitality and visual impact to the planet by adding color to it, so she first thought of a color. Can you provide the complementary color of that color?

Whalica has a mysterious permutation$$$^{\text{∗}}$$$ $$$p$$$ of length $$$n$$$. Let $$$$$$ S_k = \sum_{i = 1}^{k} p_i $$$$$$ If for all $$$1 \le i \le n$$$, $$$S_i$$$ is divisible by $$$p_i$$$, then the permutation $$$p$$$ is called a good permutation.

Checking whether a permutation is a good permutation is far too easy, and Whalica would never enjoy doing something that simple. Therefore, Whalica has specially prepared another problem for you:

Given a positive integer $$$n$$$, you need to construct a permutation $$$p$$$ of length $$$n$$$ such that it is a good permutation. You need to determine whether such a permutation exists. Moreover, if such a permutation exists, output any one permutation that satisfies the condition.

"The so-called Shichieichou (Seven-Shadow Butterflies) are the fragments of human memory..."

"The 'memories' of those who left behind attachments or regrets during their lifetime exist in this world in the form of butterflies."

"...Even though they are just memories, these are things that should not remain in this world."

"Someone must guide them back to where they truly belong."

"This is the Yamanomatsuri (Mountain Festival), which has been managed by the Sorakado family for generations."

"...There is one specific memory that I must find, no matter what..."

To fulfill the duties of the Sorakado family and to find that indispensable memory, Sorakado Ao needs to guide the Shichieichou scattered across the mountain toward the "Mayoi no Tachibana" (The Lost Mandarin Orange) during the Yamanomatsuri.

The mountain paths of Torishirojima can be represented as a tree$$$^{\text{∗}}$$$ with $$$n$$$ nodes, rooted at node $$$1$$$. It is guaranteed that every leaf node in the tree has the same depth (i.e., the distance to the root).

Due to the rugged mountain paths and the deep night, Ao's movement must follow these rules:

1. Ao starts at time $$$0$$$ from any leaf node of her choice. Her goal is to reach the root node $$$1$$$, where the "Mayoi no Tachibana" is located.
2. Ao can spend $$$1$$$ unit of time to move from her current node to its parent node.
3. During the journey, Ao may use Inari's power at most once: she can spend $$$1$$$ unit of time to take a side path and move from her current non-root node at depth $$$d$$$ to any node at depth $$$d-1$$$.
4. When Ao is at a node, she can successfully guide all Shichieichou at that location that have not yet disappeared (i.e., those with the current time $$$ \leq t_i$$$).

To allow as many Shichieichou as possible to return, please tell Ao the maximum number of butterflies she can guide.

Whalica has $$$n$$$ fans who are invited to join Whalica's lottery game.

Whalica stipulates that only non-empty strings consisting only of the characters $$$\mathtt{a, c, h, i, l, w}$$$ can be winning strings. Before the game starts, each fan writes down a valid guess string $$$s_i$$$.

There are $$$q$$$ events in total. For the $$$i$$$-th event, one of the following happens:

1. Whalica conducts a draw. Whalica provides a valid winning string $$$T_i$$$, and then reads out each character of this string from left to right. Let $$$s[l, r]$$$ denote the string formed by concatenating the characters $$$s_l, s_{l + 1}, \cdots, s_{r - 1}, s_r$$$ from $$$s$$$ in order. When Whalica reads the $$$j$$$-th character $$$T_{i,j}$$$, every fan $$$k$$$ such that $$$s_k[1, j] = T_i[1, j]$$$ (i.e., the first $$$j$$$ characters of the fan's guess string match the first $$$j$$$ characters of the winning string) receives $$$1$$$ Whalica coin. Formally, the number of Whalica coins fan $$$k$$$ gets in this draw is $$$\operatorname{LCP}(s_k, T_i)$$$, where $$$\operatorname{LCP}(s, t)$$$ denotes the length of the longest common prefix of strings $$$s, t$$$.
2. Fans modify their strings. To prevent the game from becoming monotonous, Whalica requires every fan to change their guess string. However, the fans have secretly expressed their affection for Whalica in their strings and are unwilling to make significant changes. Specifically, they do not change the length of their guess strings, but they perform one replacement on each character of their guess string, following the cyclic rule: $$$\mathtt{w \to h \to a \to l \to i \to c \to w}$$$. For example, if a fan's string is $$$\mathtt{whalica}$$$, after modification it becomes $$$\mathtt{halicwl}$$$.

Whalica is too lazy and just wants to rest. Can you help her compute how many Whalica coins she needs to pay out for each draw event?

To commemorate the eight-thousand-year wait, Iroha and Kaguya will edit an echo sequence inside the virtual space "Tsukuyomi". Whenever two echoes are perfectly in sync, the stage will light up with a flash. They want to know: how many echo sequences can make the stage flash exactly $$$k$$$ times?

More specifically, they are given an array $$$a$$$ of length $$$n$$$.

For a sequence $$$b$$$ of length $$$m$$$, define its number of flashes as $$$$$$ f\left(b\right) = \left|\left\{i | 1 \leq i \leq m - 1, b_i = b_{i+1}\right\}\right| $$$$$$ that is, the number of times two adjacent elements in the sequence are equal.

You need to compute, among the $$$2^n$$$ subsequences$$$^{\text{∗}}$$$ of $$$a$$$, how many subsequences $$$b$$$ have exactly $$$k$$$ flashes (i.e. $$$f\left(b\right) = k$$$). Since the answer may be very large, print it modulo $$$998244353$$$.

To push magicology toward a more rigorous discipline, Anisphia and Euphyllia build a mana circuit network at the detached palace. The network forms a tree$$$^{\text{∗}}$$$ with $$$n$$$ vertices labeled $$$1, 2, \cdots, n$$$, and vertex $$$i$$$ stores a mana value $$$a_i$$$.

Root the tree at vertex $$$1$$$. For a vertex $$$u$$$, let $$$\mathrm{Subtree}\left(u\right)$$$ be the set of vertices in the subtree of $$$u$$$ (including $$$u$$$ itself and all of its descendants$$$^{\text{†}}$$$).

Define one operation $$$f$$$ on an array $$$x = \left(x_1, x_2, \cdots, x_n\right)$$$ by $$$y = f\left(x\right)$$$, where $$$$$$ y_u = \sum_{v \in \mathrm{Subtree}\left(u\right)} x_v $$$$$$

They apply the operation exactly $$$k$$$ times: let $$$x^{\left(0\right)} = a$$$, and $$$x^{\left(t\right)} = f\left(x^{\left(t - 1\right)}\right)$$$ for all $$$1 \leq t \leq k$$$.

Output $$$x^{\left(k\right)}_1, x^{\left(k\right)}_2, \cdots, x^{\left(k\right)}_n$$$ modulo $$$998244353$$$.

Retinal cone cells (Cone Cells) are mainly concentrated in the fovea of the macula and are responsible for color vision and high-resolution vision. Due to differences in their light absorption spectra, their distribution ratios are significantly asymmetric:

L-cones (long-wavelength / red): about 60% – 64%.

M-cones (medium-wavelength / green): about 30% – 32%.

S-cones (short-wavelength / blue): about 2% – 10%, and they are absent in the very center of the fovea.

This distribution explains why the human eye is most sensitive to green light (covered jointly by L and M cones), while its spatial resolution for pure blue is relatively low.

yyz likes the color blue. To find the most beautiful shade of blue, she keeps collecting it using magic. For a beautiful photo, yyz will first transform it into a blue Hilbert curve.

Specifically, there are $$$n$$$ color blocks arranged from left to right. The $$$i$$$-th block contains an amount of blue element $$$b_i$$$.

When yyz starts casting magic, she will give an incantation $$$\texttt{l r x}$$$, which means: extract all blocks in the array interval $$$[l, r]$$$ that satisfy $$$b_i \geq x$$$. After a block is extracted once, it will lose all of its blue elements.

Given $$$q$$$ magic casts, please tell yyz how many blue elements can be extracted each time.

Whalica has a set $$$S$$$, and $$$S$$$ is initially empty. In addition, a parameter $$$n$$$ is given. You need to perform the given $$$q$$$ operations in order. There are four types of operations, and the format and meaning of each operation are as follows:

Operation $$$1$$$: $$$\texttt{1 x}$$$ Given an integer $$$x$$$ $$$(1\le x\le n)$$$, insert $$$x$$$ into the set $$$S$$$. In particular, if $$$S$$$ already contains the element $$$x$$$, then ignore this operation.

Operation $$$2$$$: $$$\texttt{2 x}$$$ Given an integer $$$x$$$ $$$(1\le x\le n)$$$, delete $$$x$$$ from the set $$$S$$$. In particular, if $$$S$$$ does not contain the element $$$x$$$, then ignore this operation.

Operation $$$3$$$: $$$\texttt{3}$$$ Query the smallest positive integer that does not appear in the set $$$S$$$.

Operation $$$4$$$: $$$\texttt{4}$$$ Query the weight of the set $$$S$$$. Here, the definition of the weight of the set $$$S$$$ is as follows: let the number of integers in the range $$$[1, n]$$$ that do not appear in the set $$$S$$$ be $$$k$$$, and let these $$$k$$$ elements in increasing order be $$$a_1, a_2, a_3, \cdots, a_k$$$, then the weight of the set $$$S$$$ is: $$$$$$ \sum_{i = 1}^{k} a_i \cdot 2^i $$$$$$ For each operation $$$4$$$, since the result may be very large, please take the result modulo $$$998244353$$$.

At night, there are $$$n$$$ firefly lamps lit along the riverbank. The $$$i$$$-th lamp is located at coordinate $$$x_i$$$, and they are arranged from left to right along the river, i.e., $$$$$$ x_1 \lt x_2 \lt \cdots \lt x_n $$$$$$ XqAiyee wants to walk from the $$$1$$$-st lamp to the $$$n$$$-th lamp.

When she is at the $$$i$$$-th lamp ($$$1 \le i \le n-1$$$), she can choose:

1. Stroll along the bank. Walk along the riverbank to the $$$(i+1)$$$-th lamp, consuming stamina $$$a_i$$$.
2. Fireflies lead the way. Choose an index $$$j$$$ ($$$i + 1 \le j \le n$$$), and follow the fireflies to the $$$j$$$-th lamp, consuming stamina $$$b_i + (x_j - x_i)^2$$$.

Since the fireflies will get tired, XqAiyee can use "Fireflies lead the way" at most $$$k$$$ times.

Please determine the minimum stamina required for XqAiyee to reach the $$$n$$$-th lamp.