The Solaris satellite is orbiting the Sun to collect important information about the most important celestial body for Earth. To do this, the satellite collects an array of $$$n$$$ integers and sends back $$$k$$$ checksums to ensure the accuracy of the data.

Each checksum can be represented as $$$x_1 + 2x_2 + x_3 + 2x_4 + x_5 + \cdots$$$, where $$$x$$$ is an array of integers labeled $$$x_1, x_2, x_3, \cdots$$$ that is being added up, alternating between multiplying by 1 and multiplying by 2.

More formally, let the checksum of array $$$x$$$ with length $$$l$$$ be $$$f(x)$$$ where

Given an array $$$a$$$ of $$$n$$$ integers, print out the results of $$$k$$$ checksums, where each checksum runs on a subarray of $$$a$$$ from indices $$$l_i$$$ to $$$r_i$$$ inclusive.

On Mount Olympus, gods love to send each other encoded messages. Each message consists of a ciphertext made of lowercase English letters along with $$$n$$$ phrases. To decipher the message, a Caesar$$$^\dagger$$$ cipher is applied to the ciphertext such that the $$$n$$$ phrases all appear as substrings of the resulting string.

Being the one who is always delivering these secret messages, Mercury wants to read these messages, but he cannot figure out how to decipher the text correctly. Your task is to help Mercury decipher the messages. It is guaranteed that an answer exists. If there are multiple possible answers, any valid answer will be accepted.

$$$^\dagger$$$To apply a Caesar cipher to a letter, first convert the letter to a number from 0 to 25 (A=0, B=1, C=2, ..., Z=25), and call this number $$$x$$$. When shifting by $$$k$$$, find $$$(x+k)\bmod 26$$$, and convert this number back to a letter (0=A, 1=B, 2=C, ..., 25=Z). This means that shifting Y (25) by 4 gives a result of C (3). To apply a Caesar cipher to a string, just apply the cipher for every individual letter, keeping the same $$$k$$$ each time.

Warith has been coping for a while about not being able to see Le Sserafim in Dallas, so he decided to take a mental break. He decided to go down to the Colorado River and skip some stones. When he arrived, he found $$$n$$$ stones arranged in a line. Warith knows that each stone has some mass $$$a_i$$$ $$$(1 \le i \le n)$$$. With all the stuff on his mind, he wants to maximize his enjoyment from skipping stones by picking a contiguous subarray $$$a'$$$ of the original stone array. Note that Warith cannot rearrange the stones, he can only pick a contiguous subarray from the original order.

For an array $$$a'$$$, the enjoyment from skipping all these stones is calculated as follows:

• For each unique mass $$$m$$$ in the stones of $$$a'$$$, find the number of stones with this mass ($$$f$$$)
• The enjoyment is the sum of $$$m \cdot f$$$ across all unique $$$m$$$

The only issue is that Warith also does not want the mass of the stones to vary too much either. Specifically, he doesn't want his subarray to contain more than $$$k$$$ distinct masses. With this in mind, can you help Warith maximize his enjoyment?

In the solar system, it is often said that the most habitable planet besides Earth is Mars. Indeed, scientists have recently begun settling on Mars and have set up a base for permanent living. However, while Mars has an abundance of water and soil nutrients, it's missing the greatest natural resource: Milk! Unfortunately, this means that the scientists must import milk from Earth in order to sustain themselves.

Johnny, one of the scientists, has a particular affinity for milk. He has knowledge of spaceship trips which can transport jugs of milk from Earth to Mars and vice versa. Each of these trips has a departure and arrival time, and all of these times are distinct from each other.

Due to restrictions regarding waste, in order to import milk, Johnny must place an empty jug of milk on a spaceship bound for Earth where it will be filled immediately. He can then request for a particular spaceship bound for Mars to carry the milk back, where Johnny can empty the milk jug in a large storage container. Each spaceship can only carry one of Johnny's milk jugs at most (other scientists want to use the spaceships as well!).

Before any of the spaceships begin transporting items, Johnny has two empty milk jugs on hand. Given this knowledge, Johnny would like to know the maximum jugs' worth of milk he can import from Earth with the spaceships available.

Jupiter is home to the largest Coriolis forces in the Solar System. Specifically, the Coriolis force causes horizontal bands on the planet to experience strong winds either East or West. Jupiter has $$$8$$$ of these prominent bands.

Embarking on a mission to navigate Jupiter, you are well aware of winds and need to factor them into the amount of fuel you need to bring on your mission. Your spaceship will enter Jupiter at coordinate $$$S$$$, and need to end at coordinate $$$D$$$. A coordinate can be denoted by the band you are in (vertical position), and the columns (horizontal position) along that band at time $$$t=0$$$.

Each timestep, the Coriolis winds take effect, rotating all positions of even-indexed bands right, and all positions of odd-indexed bands left.

Jupiter also has numerous storms that move along with these bands. You want to avoid these storms at all costs.

Consider the following 4 banded system example, as we can find on Earth:

        col1   col2   col3   col4   col5
band1 |   .  |   X  |   .  |   D  |   .  | <–
band2 |   X  |   .  |   X  |   X  |   X  | –>
band3 |   .  |   S  |   .  |   .  |   .  | <–
band4 |   .  |   .  |   .  |   .  |   .  | –>

After one timestep, the world would look like:

        col1   col2   col3   col4   col5
band1 |   X  |   .  |   D  |   .  |   .  | <–
band2 |   X  |   X  |   .  |   X  |   X  | –>
band3 |   S  |   .  |   .  |   .  |   .  | <–
band4 |   .  |   .  |   .  |   .  |   .  | –>

Your spaceship lets you move one tile in any cardinal direction (not diagonally) before the Coriolis effect takes effect each timestep. Note you can also choose to not move, in which you move along with the band. Also, note that columns 1 and $$$N$$$ are adjacent, but band1 and band8 are not.

Find the minimum amount of timesteps to travel from coordinate A to coordinate B (if possible) so we can know how much fuel to bring!

Note: The following problem, Uranus, may be easier if you prefer solving in order of increasing difficulty.

Christiaan Huygens was the first to propose there was a ring surrounding Saturn, in 1655. He did this by publishing his claim as an anagram: aaaaaaacccccdeeeeehiiiiiiillllmmnnnnnnnnnooooppqrrstttttuuuuu. Only three years later did he have enough evidence to publish the true claim: Annulo cingitur, tenui, plano, nusquam cohaerente, ad eclipticam inclinato ("It is encircled by a thin, flat ring, nowhere touching, inclined to the ecliptic").

Shani recently became inspired by Huygens and published some similar anagrammed claim. Unfortunately, it was mostly a joke, but now the research committee is calling upon her to back it up.

Luckily, she does have some data that she has yet to publish. Specifically, she has $$$1 \le n \le 1000$$$ data points, each a string $$$s_i$$$ with $$$1 \le |s_i| \le 100$$$. She may publish any subset of these data points, and once published, she knows the committee will concatenate them in their original order (the data points are dated). Her original claim is a string $$$t$$$ with $$$1 \le |t| \le 20$$$. The committee will then count how many times $$$t$$$ occurs as a substring in the published data (overlaps are counted), and for each occurrence she will earn one reputation score.

Report the maximum reputation score Shani can get.

Although it is not the farthest planet from the Sun, Uranus is considered to be the coldest planet in the Solar System, experiencing a minimum temperature of 49 K and extreme winds of up to 900 km/h. Undaunted, Sir Vedward V's twin brother, Sir Udward, wishes to send probes to explore Uranus.

Currently, VASA has developed $$$n$$$ probes, where the $$$i^{\text{th}}$$$ probe can survive temperatures of up to $$$x_i$$$ degrees Kelvin and wind speeds of up to $$$y_i$$$ km/h; however, it costs $$$c_i$$$ dollars to construct. In addition, VASA has conducted $$$q$$$ measurements of locations on Uranus, where the $$$i^{\text{th}}$$$ location has a temperature of $$$t_i$$$ degrees Kelvin and a wind speed of $$$w_i$$$ km/h.

Sir Udward will reward you handsomely if you help him determine, for each landing location, the cheapest probe that can survive its temperature and wind speed.