You are given $$$n$$$ points in 4-dimensional space: $$$p_1, ~p_2, ~\ldots, ~p_n$$$, where each point $$$p_i$$$ has coordinates $$$(x_i, ~y_i, ~z_i, ~w_i)$$$.

Your task is to find a point $$$o = (o_x, ~o_y, ~o_z, ~o_w)$$$ in 4-dimensional space such that the maximum distance from $$$o$$$ to any point $$$p_i$$$ is minimized. Formally, you need to minimize the value: $$$$$$ \max_{1 \leq i \leq n} \sqrt{(o_x - x_i)^2 + (o_y - y_i)^2 + (o_z - z_i)^2 + (o_w - w_i)^2} $$$$$$

Once upon a time in a land of numbers and mysteries, a mathematician named Syra discovered a magical dice. This wasn't an ordinary dice — whenever thrown, it could reveal hidden secrets of the numbers it touched. Syra wanted to test its powers, so she devised a challenge for you, the brave coder.

The magical dice is a regular six-sided die with faces numbered from 1 to 6. If you throw this dice twice, you get two outcomes — let's call them $$$a$$$ and $$$b$$$, each ranging from $$$1$$$ to $$$6$$$. Your mission is to determine whether it's possible that the sum of these two throws, $$$a + b$$$, can be equal to a given integer $$$x$$$.

You are given a set $$$S$$$ of integers, initially containing $$$n$$$ elements. Each element in $$$S$$$ is assigned a unique index from $$$1$$$ to $$$n$$$. You are also given an integer $$$p$$$ (where $$$0 \le p \le n$$$).

You will perform the following sequence of operations until the set $$$S$$$ becomes empty:

• Select an index $$$i$$$ (where $$$1 \le i \le \text{current length of }S$$$).
• Remove the element at index $$$i$$$ from $$$S$$$.
• If $$$i \gt p$$$, decrease $$$p$$$ by $$$1$$$.
• After the deletion, all elements in $$$S$$$ with indices greater than $$$i$$$ are shifted one position to the left.

The final value of $$$p$$$ after all deletions have been performed is the final score. What will be the expected final score if all operations are performed randomly?

You are given the value of $$$n$$$. For each possible initial value of $$$p$$$ (where $$$0 \le p \le n$$$), determine the expected final score after performing all deletions randomly.

You are given an array $$$a = [a_1, ~a_2, ~\ldots, ~a_n]$$$ of $$$n$$$ integers.

For each integer $$$k$$$ from $$$1$$$ to $$$n$$$, consider the following operation:

• Choose two indices $$$i$$$ and $$$j$$$ such that $$$|i - j| \ge k$$$.
• Replace $$$a_i$$$ with the bitwise OR of $$$a_i$$$ and $$$a_j$$$ ($$$a_i | a_j$$$).

You can perform this operation any number of times (possibly zero).

For each $$$k$$$ independently, determine the maximum possible value of the bitwise AND of all elements in the array after performing any number of such operations. That is, for each $$$k$$$ from $$$1$$$ to $$$n$$$, find the maximum achievable value of $$$(a_1 ~\& ~a_2 ~\& ~\ldots ~\& ~a_n)$$$.

You are given an array $$$a = [a_1, a_2, \dots, a_n]$$$ of length $$$n$$$.

An inversion is a pair of indices $$$(i, j)$$$ such that $$$1 \le i \lt j \le n$$$ and $$$a_i \gt a_j$$$. The inversion count of an array is the total number of inversions.

A cyclic shift of the first $$$k$$$ elements to the end of the array transforms $$$[a_1, a_2, \dots, a_k, a_{k+1}, \dots, a_n]$$$ into $$$[a_{k+1}, \dots, a_n, a_1, a_2, \dots, a_k]$$$.

For each $$$k = 1, ~2, ~\ldots, ~n - 1$$$, solve the following problem and print the answer.

• Find the minimum inversion count achievable in the array $$$a$$$ after performing any number(possibly zero) of cyclic shifts of the first $$$k$$$ elements to the end. For example, if $$$a = [4, 3, 5, 3, 3]$$$ and $$$k = 2$$$, we can generate these arrays $$$[4, 3, 5, 3, 3] \longrightarrow [5, 3, 3, 4, 3] \longrightarrow [3, 4, 3, 5, 3] \longrightarrow [3, 5, 3, 3, 4] \longrightarrow \ldots$$$. And the minimum inversion count among these arrays is $$$1$$$.

You are given an array $$$a_1, ~a_2, ~\ldots, ~a_n$$$ of $$$n$$$ integers. For each element $$$a_i$$$, you can perform the following operation at most once:

• Choose any set bit in the binary representation of $$$a_i$$$ and unset it (change it from 1 to 0).

Determine if it is possible to transform the array $$$a$$$ into a permutation of integers from 1 to $$$n$$$. A permutation of integers from 1 to $$$n$$$ is a sequence of $$$n$$$ distinct integers, each of which is between 1 and $$$n$$$ inclusive.

"SynergyX", a cutting-edge email service, is launching soon! To expedite the process, we need your help designing a registration system prototype.

The system operates as follows:

• Add Operation (a username): When a user registers with a username, the system checks if the username exists in the database.
  • If the username is not found, it's added to the database, and the system responds with OK.
  • If the username is found, the system generates a new username by appending numbers (starting from 1) to the original username (e.g., username1, username2, ...). It finds the smallest integer $$$i$$$ such that username$$$i$$$ is not in the database, adds username$$$i$$$ to the database, and returns username$$$i$$$ as the suggested username.
• Delete Operation (d username): When a user requests to delete their account with username, the system checks if the username exists in the database.
  • If the username is found, it's removed from the database, and the system responds with DELETED.
  • If the username is not found, the system responds with INVALID.

HR of Synergy plans to set two specific weekend days for employees for the period from a given starting date to an ending date. These weekend days will remain fixed throughout the specified period, and the objective is to maximize the total number of working days, excluding both public holidays and weekend days.

The dates should follow the Gregorian calendar, meaning they should align with the standard calendar system, including leap years. A leap year is defined as any year that:

1. Is evenly divisible by 4, and
2. Is not evenly divisible by 100, unless it is also evenly divisible by 400.

For example:

• 2024 is a leap year (divisible by 4 but not by 100).
• 1900 is not a leap year (divisible by 100 but not by 400).
• 2000 is a leap year (divisible by 400).

In leap years, February has 29 days instead of the usual 28.

A working day is any day that is neither a public holiday nor a designated weekend day. The HR wants to select two specific days of the week that will serve as weekend days. Your task is to help him choose these two days such that the total number of working days is maximized.